Write the sum $LaTeX: \displaystyle -21-29-37 \ldots -341-349$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-29-(-21)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -21+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -21+(n-1)(-8)=-349 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(- 8 n - 13\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-21-349) }{2}=-7770$ .