Write the sum $LaTeX: \displaystyle 24+33+42 \ldots +420+429$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=33-(24)=9$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= 24+(n-1)(9)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle 24+(n-1)(9)=429 \implies n = 46$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{46} \left(9 n + 15\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 46(24+429) }{2}=10419$ .