Write the sum $LaTeX: \displaystyle 16+25+34 \ldots +196+205$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=25-(16)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 16+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 16+(n-1)(9)=205 \implies n = 22$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{22} \left(9 n + 7\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 22(16+205) }{2}=2431$ .