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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:
\[3\left( - 1 \right) + 7\left( - 1 \right) + 11\left( - 1 \right) + . . . + \left( 4n - 1 \right) \left( - 1 \right)\]
\[= - \left[ 3 + 7 + 11 + . . . + \left( 4n - 1 \right) \right]\]
\[ = - \left[ \frac{n}{2}\left\{ 3 \times 2 + \left( n - 1 \right)4 \right\} \right]\]
\[ = - \left[ \frac{n}{2}\left( 4n + 2 \right) \right]\]
\[ = - n\left( 2n + 1 \right)\]
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