Write the sum $LaTeX: \displaystyle 6+7+8 \ldots +36+37$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=7-(6)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 6+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 6+(n-1)(1)=37 \implies n = 32$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{32} \left(n + 5\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 32(6+37) }{2}=688$ .