Write the sum $LaTeX: \displaystyle 45+52+59 \ldots +199+206$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=52-(45)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 45+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 45+(n-1)(7)=206 \implies n = 24$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{24} \left(7 n + 38\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 24(45+206) }{2}=3012$ .