Write the sum $LaTeX: \displaystyle -49-42-35 \ldots +147+154$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-42-(-49)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -49+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -49+(n-1)(7)=154 \implies n = 30$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{30} \left(7 n - 56\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 30(-49+154) }{2}=1575$ .