qu.1.topic=NA Series@

qu.1.1.question=<div title="University of Waterloo Statistics Bank/Numerical Analysis/Sequences and Series/Series/Q$Q">The sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math> is:</div>@
qu.1.1.answer.num=$Ans@
qu.1.1.answer.units=@
qu.1.1.showUnits=false@
qu.1.1.grading=exact_value@
qu.1.1.negStyle=minus@
qu.1.1.numStyle=thousands scientific dollars arithmetic@
qu.1.1.mode=Numeric@
qu.1.1.name=04. Sum of squares@
qu.1.1.comment=<p>Use the formula <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>k</mi><mfenced open='(' close=')' separators=','><mrow><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></mstyle></math> and some basic series algebra to rewrite this sum as a difference of two known sums:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$mm1</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$np1</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Twonp1</mi></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi mathvariant='normal'>$mm1</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$m</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Twomm1p1</mi></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Sum1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Sum2</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$Q=4;
$n=range(15,30,1);
$m=range(3,12,1);
$mm1=$m-1;
$np1=$n+1;
$Twonp1=2*$n+1;
$Twomm1p1=2*$m-1;
$Sum1=$n*($n+1)*(2*$n+1)/6;
$Sum2=($m-1)*$m*(2*$m-1)/6;
$Ans=$Sum1-$Sum2;@
qu.1.1.uid=a11de3b8-8825-411c-9310-74a88e2e5d71@
qu.1.1.info=  Type=numeric;
  Course=230;
@

qu.1.2.question=<div title="University of Waterloo Statistics Bank/Numerical Analysis/Sequences and Series/Series/Q$Q">The value of the sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><mi>k</mi></mrow></mfenced></mrow></mstyle></math> is:</div>@
qu.1.2.answer.num=$Ans@
qu.1.2.answer.units=@
qu.1.2.showUnits=false@
qu.1.2.grading=exact_value@
qu.1.2.negStyle=minus@
qu.1.2.numStyle=thousands scientific dollars arithmetic@
qu.1.2.mode=Numeric@
qu.1.2.name=01a. Sum of a+b<i>k</i>@
qu.1.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><mi>k</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi mathvariant='normal'>$bd</mi><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>k</mi></mrow></mstyle></math> <br />
= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$an</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$Q="01a";
$a=range(1,8);
$b=range(1,8);
$bd=if(eq($b,1),"",$b);
$n=range(10,30,5);
$an=$a*$n;
$Ans=$n*$a+$b*($n)*($n+1)/2;
$Alt1=range(45,$Ans-15,5);
$Alt2=$Ans+range(20,60,2);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.2.uid=cb39df44-1986-47c5-9ce3-b5c5ca736a4c@
qu.1.2.info=  Type=numeric;
  Course=230;
@

qu.1.3.question=<div title="University of Waterloo Statistics Bank/Numerical Analysis/Sequences and Series/Series/Q$Q">Evaluate the sum:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ad</mi><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math></div>@
qu.1.3.answer.num=$Ans@
qu.1.3.answer.units=@
qu.1.3.showUnits=false@
qu.1.3.grading=exact_value@
qu.1.3.negStyle=minus@
qu.1.3.numStyle=thousands scientific dollars arithmetic@
qu.1.3.mode=Numeric@
qu.1.3.name=06. Sum a quadratic II@
qu.1.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ad</mi><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ad</mi><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ad</mi><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2d</mi><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mi mathvariant='normal'>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2d</mi><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mi mathvariant='normal'>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a2d</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2d</mi></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mn>1</mn></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2d</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></munderover><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2d</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></munderover><mn>1</mn></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a2d</mi><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2d</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$n1</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$n1</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a2d</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SOSTerm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SumTerm</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2d</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SOSTerm1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SumTerm1</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n1</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sum</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$Sum1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$Q=6;
$m=range(3,12);
$n=range($m+3,$m+12);
$n1=$m-1;
$a=range(1,4,1)*-1^rint(2);
$ad=if(eq($a,-1),"-",if(eq($a,1),"",$a));
$b=range(1,5);
$a2=$a^2;
$a2d=if(eq($a2,1),"",$a2);
$b2=$b^2;
$b2d=if(eq($b2,1),"",$b2);
$abT2=abs(2*$a*$b);
$Sign=if(gt($a,0),"-","+");
$SOSTerm=$n*($n+1)*(2*$n+1)/6;
$SOSTerm1=$n1*($n1+1)*(2*$n1+1)/6;
$SumTerm=$n*($n+1)/2;
$SumTerm1=$n1*($n1+1)/2;
$Sum=$a2*$SOSTerm - 2*$a*$b*$SumTerm + $b2*$n;
$Sum1=$a2*$SOSTerm1 - 2*$a*$b*$SumTerm1 + $b2*$n1;
$Ans=$Sum-$Sum1;@
qu.1.3.uid=5bbfe89f-3aca-499d-ade0-70a843fcca7f@
qu.1.3.info=  Type=numeric;
  Difficulty=4;
  Course=230;
@

qu.1.4.question=<div title="University of Waterloo Statistics Bank/Numerical Analysis/Sequences and Series/Series/Q$Q">Evaluate the sum:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ad</mi><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></div>@
qu.1.4.answer.num=$Ans@
qu.1.4.answer.units=@
qu.1.4.showUnits=false@
qu.1.4.grading=exact_value@
qu.1.4.negStyle=minus@
qu.1.4.numStyle=thousands scientific dollars arithmetic@
qu.1.4.mode=Numeric@
qu.1.4.name=05. Sum a quadratic (start \@ 1)@
qu.1.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ad</mi><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2d</mi><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a2d</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><msup><mi>k</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2d</mi></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mn>1</mn></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a2d</mi><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>6</mn></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi></mrow></mstyle></math><br />
<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SOSTerm</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Sign</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$abT2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$SumTerm</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$Q=5;
$m=1;
$n=range($m+2,$m+12);
$a=range(1,4)*-1^rint(2);
$ad=if(eq($a,1),"",if(eq($a,-1),"-",$a));
$b=range(1,5);
$a2=$a^2;
$a2d=if(eq($a2,1),"",$a2);
$b2=$b^2;
$b2d=if(eq($b2,1),"",$b2);
$abT2=abs(2*$a*$b);
$Sign=if(gt(2*$a*$b,0),"-","+");
$SOSTerm=$n*($n+1)*(2*$n+1)/6;
$SumTerm=$n*($n+1)/2;
$Ans=$a2*$SOSTerm $Sign $abT2*$SumTerm + $b2*$n;@
qu.1.4.uid=b02363f3-961d-4881-ad26-62f911c46715@
qu.1.4.info=  Type=numeric;
  Difficulty=3;
  Course=230;
@

qu.1.5.mode=Multiple Choice@
qu.1.5.name=02. Sum of integers from <i>m</i> to <i>n</i>@
qu.1.5.comment=<p>The trick when the sums look familar but the indices are wrong is to rewrite the sum as a combination of two (or more) sums that do fit the patterm needed to use a known sum. Here we want to use the known result <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000' veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>n</mi><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>k</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></munderover><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mi mathvariant='normal'>$m</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.5.editing=useHTML@
qu.1.5.solution=@
qu.1.5.algorithm=$Q=2;
$n=range(25,40,1);
$m=range(3,8,1);
$Ans=$n*($n+1)/2 - ($m-1)*$m/2;
$Alt1=int(range(0.25*$Ans,0.75*$Ans,2));
$Alt2=int(range(1.1*$Ans,1.4*$Ans,1));
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.5.uid=43631eb2-17b3-455c-b2f2-ed837847fa71@
qu.1.5.info=  Type=MC;
  Course=230;
@
qu.1.5.question=<div title="University of Waterloo Statistics Bank/Numerical Analysis/Sequences and Series/Series/Q$Q">The sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$m</mi></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>k</mi></mrow></mstyle></math> is</div>@
qu.1.5.answer=1@
qu.1.5.choice.1=$Ans@
qu.1.5.choice.2=$Alt1@
qu.1.5.choice.3=$Alt2@
qu.1.5.choice.4=$Alt3@
qu.1.5.fixed=@

qu.1.6.question=<div title="University of Waterloo Statistics Bank/Numerical Analysis/Sequences and Series/Series/Q$Q"><img hspace="4" align="$Align" title="Beer [IMG:Beer$Which.gif]" src="__BASE_URI__NA/SS/Series/Beer$Which.gif" alt="Beer" />Every year since I turned 19 my father has given me as many bottles of beer as my age (19 then 20 then ...). Unfortunately, I don't drink so that beer is just stored in my garage. If I just turned $Age years old, how many bottles of Dad's beer do I have?</div>@
qu.1.6.answer.num=$Ans@
qu.1.6.answer.units=@
qu.1.6.showUnits=false@
qu.1.6.grading=exact_value@
qu.1.6.negStyle=minus@
qu.1.6.numStyle=thousands scientific dollars arithmetic@
qu.1.6.mode=Numeric@
qu.1.6.name=03. Bottles of beer@
qu.1.6.comment=<p>Let k represent your age. Then the number of bottles received is a simple sum:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>19</mn></mrow><mrow><mi mathvariant='normal'>$Age</mi></mrow></munderover><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$Age</mi></mrow></munderover><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mn>18</mn></mrow></munderover><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$Age</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$Age</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mn>18</mn><mfenced open='(' close=')' separators=','><mrow><mn>19</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.6.editing=useHTML@
qu.1.6.solution=@
qu.1.6.algorithm=$Q=3;
$Which=rint(4);
$Align=switch(rint(2),"Left","Right");
$Age=range(30,62);
$Ans=$Age*($Age+1)/2 - 9*19;@
qu.1.6.uid=5fc8068b-7238-47f7-a666-0c91df4c9b3e@
qu.1.6.info=  Type=numeric;
  Course=230;
@

qu.1.7.mode=Multiple Choice@
qu.1.7.name=01b.  Sum of a+b<i>k</i>@
qu.1.7.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><mi>k</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi mathvariant='normal'>$bd</mi><mi>k</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mi>k</mi></mrow></mstyle></math> <br />
= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$an</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$n</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$Ans</mi></mrow></mstyle></math></p>@
qu.1.7.editing=useHTML@
qu.1.7.solution=@
qu.1.7.algorithm=$Q="01b";
$a=range(1,8);
$b=range(1,8);
$bd=if(eq($b,1),"",$b);
$n=range(10,30,5);
$an=$a*$n;
$Ans=$n*$a+$b*($n)*($n+1)/2;
$Alt1=range(45,$Ans-15,5);
$Alt2=$Ans+range(20,60,2);
$Alt3=int(0.5*($Ans+switch(rint(2),$Alt1,$Alt2)));@
qu.1.7.uid=e7d450f8-0009-41a6-9a30-49d8e8d56190@
qu.1.7.info=  Type=MC;
  Course=230;
@
qu.1.7.question=<div title="University of Waterloo Statistics Bank/Numerical Analysis/Sequences and Series/Series/Q$Q">The value of the sum <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.1666667em' stretchy='true' largeop='true' movablelimits='true'>&Sum;</mo><mrow><mi>k</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></munderover><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bd</mi><mi>k</mi></mrow></mfenced></mrow></mstyle></math> is:</div>@
qu.1.7.answer=1@
qu.1.7.choice.1=$Ans@
qu.1.7.choice.2=$Alt1@
qu.1.7.choice.3=$Alt2@
qu.1.7.choice.4=$Alt3@
qu.1.7.fixed=@

