Write the sum $LaTeX: \displaystyle 29+37+45 \ldots +213+221$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=37-(29)=8$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 29+(n-1)(8)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 29+(n-1)(8)=221 \implies n = 25$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{25} \left(8 n + 21\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 25(29+221) }{2}=3125$ .