Write the sum $LaTeX: \displaystyle 47+51+55 \ldots +155+159$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=51-(47)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 47+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 47+(n-1)(4)=159 \implies n = 29$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{29} \left(4 n + 43\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 29(47+159) }{2}=2987$ .