Write the sum $LaTeX: \displaystyle 21+22+23 \ldots +57+58$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=22-(21)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 21+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 21+(n-1)(1)=58 \implies n = 38$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{38} \left(n + 20\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 38(21+58) }{2}=1501$ .