Summation

Jul 2008
177
4
Show that \(\displaystyle \sum (-1)^{r+1} r^2 \) = -n(2n+1)


summation is from r=1 to 2n . sry i dont know how to type

Thx:)
 
Sep 2008
1,261
539
West Malaysia
Show that \(\displaystyle \sum (-1)^{r+1} r^2 \) = -n(2n+1)


summation is from r=1 to 2n . sry i dont know how to type

Thx:)
hi

\sum^{2n}_{r=1}(-1)^{r+1} r^2 will generate

\(\displaystyle \sum^{2n}_{r=1}(-1)^{r+1} r^2\)

\(\displaystyle =1-4+9-16+25-36+...\)

\(\displaystyle =1^2-2^2+3^2-4^2+...+(2n-1)^2-(2n)^2\)

\(\displaystyle =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n-1)\)

\(\displaystyle =-3-7-...-(4n-1)\)

which is an AP with first term -3 , last term -(4n-1)

\(\displaystyle S_n=\frac{n}{2}(-3-(4n-1))\)

simplify this and you are done.
 
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Oct 2009
255
20
St. Louis Area
hi

\sum^{2n}_{r=1}(-1)^{r+1} r^2 will generate

\(\displaystyle \sum^{2n}_{r=1}(-1)^{r+1} r^2\)

\(\displaystyle =1-4+9-16+25-36+...\)

\(\displaystyle =1^2-2^2+3^2-4^2+...+(2n)^2-(2n-1)^2\)

\(\displaystyle =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)\)

\(\displaystyle =-3-4-...-(4n-1)\)

which is an AP with first term -3 , last term -(4n-1)

\(\displaystyle S_n=\frac{n}{2}(-3-(4n-1))\)

simplify this and you are done.
I didn't quite follow this.

How did you get \(\displaystyle =1^2-2^2+3^2-4^2+...+(2n)^2-(2n-1)^2\)?

When I do this I come up with \(\displaystyle =1^2-2^2+3^2-4^2+...-(2n)^2 +(2n-1)^2\), because \(\displaystyle (-1)^{2n+1} {2n}^2\) = \(\displaystyle -{2n}^2\).
 
Jul 2008
177
4
I didn't quite follow this.

How did you get \(\displaystyle =1^2-2^2+3^2-4^2+...+(2n)^2-(2n-1)^2\)?

When I do this I come up with \(\displaystyle =1^2-2^2+3^2-4^2+...-(2n)^2 +(2n-1)^2\), because \(\displaystyle (-1)^{2n+1} {2n}^2\) = \(\displaystyle -{2n}^2\).
yep i realise i dont get it. i agree with oldguynewstudnet.and also in the part \(\displaystyle
=(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)
\)


it shd be -3-7 and not -3-4
 
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Sep 2008
1,261
539
West Malaysia
yep i realise i dont get it. i agree with oldguynewstudnet.and also in the part \(\displaystyle
=(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)
\)

it shd be -3-7 and not -3-4
yup , some algebraic error here and there . Edited .