Write the sum $LaTeX: \displaystyle 50+53+56 \ldots +170+173$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=53-(50)=3$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 50+(n-1)(3)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 50+(n-1)(3)=173 \implies n = 42$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{42} \left(3 n + 47\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 42(50+173) }{2}=4683$ .