sum of non-arithmetic,non-geometric series

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March 8th 2011, 03:37 AM
chris86

sum of non-arithmetic,non-geometric series

Find the sum of the series:
(a) (1^2)+(3^2)+(5^2)+...+(2n-1)^2
(b) (1^2)-(2^2)+(3^2)-(4^2)+...+((-1)^(n+1))(n^2)
March 8th 2011, 03:48 AM
mr fantastic

Quote:

Originally Posted by chris86

Find the sum of the series:
(a) (1^2)+(3^2)+(5^2)+...+(2n-1)^2
(b) (1^2)-(2^2)+(3^2)-(4^2)+...+((-1)^(n+1))(n^2)

Sum of the Squares of the first n Natural Numbers
March 8th 2011, 04:31 AM
chisigma

Quote:

Originally Posted by chris86

Find the sum of the series:
(a) (1^2)+(3^2)+(5^2)+...+(2n-1)^2
(b) (1^2)-(2^2)+(3^2)-(4^2)+...+((-1)^(n+1))(n^2)

a)

$\displaystyle 1^{2} + 3^{2} + 5^{2} + ...+ (2n-1)^{2} =$

$\displaystyle = 1^{2} + 2^{2} + 3^{2} + ...+ (2n-1)^{2} - 2^{2}\ \{1^{2} + 2^{2} + 3^{2} + ...+ (n-1)^{2}\} =$

$\displaystyle = \frac{2n\ (2n-1)\ (4n-1) - 4n\ (n-1)\ (2n-1)}{6} =$

$\displaystyle = \frac{2n\ (2n-1)\ (2n+1)}{6}$

Kind regards

$\chi$ $\sigma$

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