Write the sum $LaTeX: \displaystyle -4-9-14 \ldots -224-229$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-9-(-4)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -4+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -4+(n-1)(-5)=-229 \implies n = 46$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{46} \left(1 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 46(-4-229) }{2}=-5359$ .