Write the sum $LaTeX: \displaystyle 0+7+14 \ldots +196+203$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=7-(0)=7$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 0+(n-1)(7)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 0+(n-1)(7)=203 \implies n = 30$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{30} \left(7 n - 7\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 30(0+203) }{2}=3045$ .