Write the sum $LaTeX: \displaystyle -10-6-2 \ldots +110+114$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-6-(-10)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -10+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -10+(n-1)(4)=114 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(4 n - 14\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(-10+114) }{2}=1664$ .