Write the sum $LaTeX: \displaystyle -41-36-31 \ldots +59+64$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-36-(-41)=5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -41+(n-1)(5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -41+(n-1)(5)=64 \implies n = 22$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{22} \left(5 n - 46\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 22(-41+64) }{2}=253$ .