Write the sum $LaTeX: \displaystyle -7-5-3 \ldots +91+93$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-5-(-7)=2$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -7+(n-1)(2)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -7+(n-1)(2)=93 \implies n = 51$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{51} \left(2 n - 9\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 51(-7+93) }{2}=2193$ .