Write the sum $LaTeX: \displaystyle 30+23+16 \ldots -138-145$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=23-(30)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 30+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 30+(n-1)(-7)=-145 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(37 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(30-145) }{2}=-1495$ .