Write the sum $LaTeX: \displaystyle 17+9+1 \ldots -303-311$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=9-(17)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 17+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 17+(n-1)(-8)=-311 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(25 - 8 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(17-311) }{2}=-6174$ .