Summation

**helloying** · May 22nd 2010, 10:43 PM

Show that $\sum (-1)^{r+1} r^2$ = -n(2n+1)

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

Thx

**mathaddict** · May 23rd 2010, 04:25 AM

Originally Posted by helloying

Show that $\sum (-1)^{r+1} r^2$ = -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

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

$=1-4+9-16+25-36+...$

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

$=(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n-1)$

$=-3-7-...-(4n-1)$

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

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

simplify this and you are done.

**oldguynewstudent** · May 23rd 2010, 07:10 AM

Originally Posted by mathaddict

hi

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

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

$=1-4+9-16+25-36+...$

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

$=(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)$

$=-3-4-...-(4n-1)$

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

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

simplify this and you are done.

I didn't quite follow this.

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

When I do this I come up with $=1^2-2^2+3^2-4^2+...-(2n)^2 +(2n-1)^2$ , because $(-1)^{2n+1} {2n}^2$ = $-{2n}^2$ .

**helloying** · May 24th 2010, 01:06 AM

Originally Posted by oldguynewstudent

I didn't quite follow this.

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

When I do this I come up with $=1^2-2^2+3^2-4^2+...-(2n)^2 +(2n-1)^2$ , because $(-1)^{2n+1} {2n}^2$ = $-{2n}^2$ .

yep i realise i dont get it. i agree with oldguynewstudnet.and also in the part $<br /> =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)<br />$

it shd be -3-7 and not -3-4

**mathaddict** · May 27th 2010, 11:38 PM

Originally Posted by helloying

yep i realise i dont get it. i agree with oldguynewstudnet.and also in the part $<br /> <font color=$ =(1+2)(1-2)+(3+4)(3-4)+...+(2n+2n-1)(2n-2n+1)
" alt="
=(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 .

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