Question

Find the fifteenth term of a sequence where the tenth term is 17 and the common difference is seven. Give the formula for the general term.

Short answer

The fifteenth term is 144 and the general term is$a_n=7n+39$.

Step-by-step solution

Step 1. Given Information.

The tenth term is 17 and the common difference is seven.

Step 2. Find the first term by using the arithmetic formula.

Substitute 10 for n, 7 for d, and the tenth term is 17, in the formula $a_n=a_1+\left(n-1\right)d$.

$\begin{array}{rcl}{a}_{10}& =& a_1+\left(10-1\right)7\\ 17& =& a_1+63\\ a_1& =& 63-17\\ & =& 46\end{array}$

Step 3. Find the fifteenth term.  

Use the first term and the common difference to find the fifteenth term.

$\begin{array}{rcl}{a}_{15}& =& 46+\left(15-1\right)7\\ & =& 46+98\\ & =& 144\end{array}$

Step 4. Find the general terms.

To find the general terms, substitute the first term 46 and the common difference 7 in the formula.

$\begin{array}{rcl}{a}_n& =& 46+\left(n-1\right)7\\ & =& 46+7n-7\\ & =& 7n+39\end{array}$