Chapter 5 - Number Theory and the Real Number System - 5.7 Arithmetic and Geometric Sequences - Exercise Set 5.7 - Page 330: 116

The sequence is arithmetic and the sum of the first $10$ terms is $610$.

The terms have a common difference of $12$ so the sequence is arithmetic with $d=12$ and $a_1=7$. The sum $S_n$ of the first $n$ terms of an arithmetic sequence is given by the formula: $$S_n=\frac{n}{2}(a_1+a_n)$$ The given sequence has $a_1=7$ and $d=12$. However, the value of $a_{10}$ (the tenth term) is not yet known. Solve for $a_{10}$ using the formula $a_n=a_1+(n−1)(d)$ to obtain: \begin{align*} a_{10}&=7+(10−1)(12)\\ &=7+9(12)\\ &=7+108\\ &=115 \end{align*} Use the formula for the sum of the first $n$ terms to obtain the sum of the first $10$ terms (use $n=10$) to obtain: \begin{align*} S_{10}&=\frac{10}{2}(7+115)\\ &=5(122)\\ &=610 \end{align*}