Write the sum $LaTeX: \displaystyle 19+27+35 \ldots +179+187$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=27-(19)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 19+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 19+(n-1)(8)=187 \implies n = 22$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{22} \left(8 n + 11\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 22(19+187) }{2}=2266$ .