Write the sum $LaTeX: \displaystyle 34+41+48 \ldots +377+384$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=41-(34)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 34+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 34+(n-1)(7)=384 \implies n = 51$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{51} \left(7 n + 27\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 51(34+384) }{2}=10659$ .