Write the sum $LaTeX: \displaystyle 11+12+13 \ldots +31+32$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=12-(11)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 11+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 11+(n-1)(1)=32 \implies n = 22$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{22} \left(n + 10\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 22(11+32) }{2}=473$ .