Write the sum $LaTeX: \displaystyle 18+23+28 \ldots +178+183$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=23-(18)=5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 18+(n-1)(5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 18+(n-1)(5)=183 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(5 n + 13\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(18+183) }{2}=3417$ .