Write the sum $LaTeX: \displaystyle -21-17-13 \ldots +139+143$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-17-(-21)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -21+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -21+(n-1)(4)=143 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(4 n - 25\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-21+143) }{2}=2562$ .