Write the sum $LaTeX: \displaystyle -3+0+3 \ldots +117+120$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=0-(-3)=3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -3+(n-1)(3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -3+(n-1)(3)=120 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(3 n - 6\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(-3+120) }{2}=2457$ .