Write the sum $LaTeX: \displaystyle 13+14+15 \ldots +56+57$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=14-(13)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 13+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 13+(n-1)(1)=57 \implies n = 45$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{45} \left(n + 12\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 45(13+57) }{2}=1575$ .