Write the sum $LaTeX: \displaystyle -27-19-11 \ldots +245+253$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-19-(-27)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -27+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -27+(n-1)(8)=253 \implies n = 36$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{36} \left(8 n - 35\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 36(-27+253) }{2}=4068$ .