Write the sum $LaTeX: \displaystyle 48+53+58 \ldots +183+188$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=53-(48)=5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 48+(n-1)(5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 48+(n-1)(5)=188 \implies n = 29$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{29} \left(5 n + 43\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 29(48+188) }{2}=3422$ .