Answer
$\sum\limits_{k=3}^{n}{{{\left( -1 \right)}^{k+1}}{{k}^{2}}}$.
Work Step by Step
$9-16+25+\ldots +{{\left( -1 \right)}^{n+1}}{{n}^{2}}$
This is the sum of the square of natural numbers, and the sign alternates.
Thus, the sigma notation is,
$\sum\limits_{k=3}^{n}{{{\left( -1 \right)}^{k+1}}{{k}^{2}}}$
Thus, the sigma notation for the sum $9-16+25+\ldots +{{\left( -1 \right)}^{n+1}}{{n}^{2}}$ is $\sum\limits_{k=3}^{n}{{{\left( -1 \right)}^{k+1}}{{k}^{2}}}$.
