Write the sum $LaTeX: \displaystyle -4+5+14 \ldots +437+446$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=5-(-4)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -4+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -4+(n-1)(9)=446 \implies n = 51$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{51} \left(9 n - 13\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 51(-4+446) }{2}=11271$ .