Write the sum $LaTeX: \displaystyle 44+53+62 \ldots +296+305$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=53-(44)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 44+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 44+(n-1)(9)=305 \implies n = 30$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{30} \left(9 n + 35\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 30(44+305) }{2}=5235$ .