Write the sum $LaTeX: \displaystyle 7+11+15 \ldots +195+199$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=11-(7)=4$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 7+(n-1)(4)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 7+(n-1)(4)=199 \implies n = 49$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{49} \left(4 n + 3\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 49(7+199) }{2}=5047$ .