1^2 −3^2 +5^2 −7^2 +•••+17^2 −19^2


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Question Number 30245 by .none. last updated on 19/Feb/18

$1^{2} - 3^{2} + 5^{2} - 7^{2} + ∙ ∙ ∙ + 17^{2} - 19^{2}$
Commented by Penguin last updated on 19/Feb/18

$= \underset{n = 1}{\sum^{10}} {(- 1)}^{n - 1} {(2 n - 1)}^{2}$
Commented by abdo imad last updated on 20/Feb/18
$let put S_N (x)=Σ_(n=1) ^N (2n−1)^2 x^n and xfrom R −{1} S_N (x)=Σ_(n=1) ^N (4n^2 −4n +1)x^n =4Σ_(n=1) ^N n^2 x^n −4Σ_(n=) ^N n x^n + Σ_(n=1) ^N x^n =4A_n (x) −4B_n (x) +C_n (x) we have C_n (x)=x Σ_(n=1) ^N x^(n−1) =xΣ_(n=0) ^(N−1) x^n =x ((1−x^N )/(1−x)) ⇒((dC_n (x))/dx)= (d/dx)(((x−x^(N+1) )/(1−x)))=Σ_(n=1) ^N nx^(n−1) =((B_n (x))/x) ⇒ B_n (x)=x (d/dx)( ((x −x^(N+1) )/(1−x)))=x (((1−(N+1)x^N +(x−x^(N+1) ))/((1−x^ )^2 ))... also ((dB_n (x))/dx)=Σ_(n=1) ^N n^2 x^(n−1) =(1/x)Σ_(n=1) ^N n^2 x^n ⇒ A_n (x)=x ((dB_n (x))/dx) ...after finding A_n ,B_n ,C_n take x=−1 and N=20 ....$
$l e t p u t S_{N} (x) = \sum_{n = 1}^{N} {(2 n - 1)}^{2} x^{n} a n d x f r o m R - {1}$ $S_{N} (x) = \sum_{n = 1}^{N} (4 n^{2} - 4 n + 1) x^{n}$ $= 4 \sum_{n = 1}^{N} n^{2} x^{n} - 4 \sum_{n =}^{N} n x^{n} + \sum_{n = 1}^{N} x^{n}$ $= 4 A_{n} (x) - 4 B_{n} (x) + C_{n} (x) w e h a v e$ $C_{n} (x) = x \sum_{n = 1}^{N} x^{n - 1} = x \sum_{n = 0}^{N - 1} x^{n} = x \frac{1 - x^{N}}{1 - x}$ $\Rightarrow \frac{{d C}_{n} (x)}{d x} = \frac{d}{d x} (\frac{x - x^{N + 1}}{1 - x}) = \sum_{n = 1}^{N} {n x}^{n - 1} = \frac{B_{n} (x)}{x} \Rightarrow$ $B_{n} (x) = x \frac{d}{d x} (\frac{x - x^{N + 1}}{1 - x}) = x \frac{(1 - (N + 1) x^{N} + (x - x^{N + 1})}{{(1 - x^{})}^{2}} . . .$ $a l s o \frac{{d B}_{n} (x)}{d x} = \sum_{n = 1}^{N} n^{2} x^{n - 1} = \frac{1}{x} \sum_{n = 1}^{N} n^{2} x^{n} \Rightarrow$ $A_{n} (x) = x \frac{{d B}_{n} (x)}{d x} . . . a f t e r f i n d i n g A_{n}, B_{n}, C_{n} t a k e x = - 1$ $a n d N = 20 . . . .$
Commented by abdo imad last updated on 20/Feb/18

$N = 10 .$
	Answered by ajfour last updated on 20/Feb/18

	$= (1 - 3) (1 + 3) + (5 - 7) (5 + 7) + . .$ $. . . + (17 - 19) (17 + 19)$ $= - 2 (1 + 3 + 5 + 7 + . . . . . + 17 + 19)$ $= - 2 \times 10^{2} = - 200 .$
	Commented by abdo imad last updated on 20/Feb/18

	$r e a l l y y o u h a v e a g o o d l o o k i n g f o r t h i s q u s t i o n t h a n k s . . .$
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