Write the sum $LaTeX: \displaystyle 48+41+34 \ldots -176-183$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=41-(48)=-7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 48+(n-1)(-7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 48+(n-1)(-7)=-183 \implies n = 34$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{34} \left(55 - 7 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 34(48-183) }{2}=-2295$ .