Write the sum $LaTeX: \displaystyle -13-22-31 \ldots -220-229$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-22-(-13)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -13+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -13+(n-1)(-9)=-229 \implies n = 25$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{25} \left(- 9 n - 4\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 25(-13-229) }{2}=-3025$ .