Write the sum $LaTeX: \displaystyle 27+34+41 \ldots +160+167$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=34-(27)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 27+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 27+(n-1)(7)=167 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(7 n + 20\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(27+167) }{2}=2037$ .