Write the sum $LaTeX: \displaystyle 49+44+39 \ldots -46-51$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=44-(49)=-5$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 49+(n-1)(-5)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 49+(n-1)(-5)=-51 \implies n = 21$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{21} \left(54 - 5 n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 21(49-51) }{2}=-21$ .