Write the sum $LaTeX: \displaystyle 0+1+2 \ldots +31+32$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=1-(0)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 0+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 0+(n-1)(1)=32 \implies n = 33$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{33} \left(n - 1\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 33(0+32) }{2}=528$ .