Write the sum $LaTeX: \displaystyle -43-46-49 \ldots -151-154$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-46-(-43)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -43+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -43+(n-1)(-3)=-154 \implies n = 38$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{38} \left(- 3 n - 40\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 38(-43-154) }{2}=-3743$ .