Write the sum $LaTeX: \displaystyle 45+42+39 \ldots -69-72$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=42-(45)=-3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 45+(n-1)(-3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 45+(n-1)(-3)=-72 \implies n = 40$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{40} \left(48 - 3 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 40(45-72) }{2}=-540$ .