Question

Write the sum of the series 1² − 2² + 3² − 4² + 5² − 6² + ... + (2n − 1)² − (2n)².

Answer 1

The given series can be rewritten as:

\[3\left( - 1 \right) + 7\left( - 1 \right) + 11\left( - 1 \right) + . . . + \left( 4n - 1 \right) \left( - 1 \right)\]

\[= - \left[ 3 + 7 + 11 + . . . + \left( 4n - 1 \right) \right]\]

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

\[ = - \left[ \frac{n}{2}\left( 4n + 2 \right) \right]\]

\[ = - n\left( 2n + 1 \right)\]