Write the sum $LaTeX: \displaystyle 31+35+39 \ldots +191+195$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=35-(31)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 31+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 31+(n-1)(4)=195 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(4 n + 27\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(31+195) }{2}=4746$ .