Write the sum $LaTeX: \displaystyle 29+28+27 \ldots -7-8$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=28-(29)=-1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 29+(n-1)(-1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 29+(n-1)(-1)=-8 \implies n = 38$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{38} \left(30 - n\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 38(29-8) }{2}=399$ .