Write the sum $LaTeX: \displaystyle 49+41+33 \ldots -335-343$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=41-(49)=-8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 49+(n-1)(-8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 49+(n-1)(-8)=-343 \implies n = 50$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{50} \left(57 - 8 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 50(49-343) }{2}=-7350$ .