Write the sum $LaTeX: \displaystyle -41-50-59 \ldots -401-410$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-50-(-41)=-9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -41+(n-1)(-9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -41+(n-1)(-9)=-410 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(- 9 n - 32\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-41-410) }{2}=-9471$ .