Question:
Write the sum of the series 12 − 22 + 32 − 42 + 52 − 62 + ... + (2n − 1)2 − (2n)2.
Solution:
The given series can be rewritten as:
$3(-1)+7(-1)+11(-1)+\ldots+(4 n-1)(-1)$
$=-[3+7+11+\ldots+(4 n-1)]$
$=-\left[\frac{n}{2}\{3 \times 2+(n-1) 4\}\right]$
$=-\left[\frac{n}{2}(4 n+2)\right]$
$=-n(2 n+1)$
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