Question

Find the first term and common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the formula for the general term.

Short answer

The first term, common difference, and the general formula are 5, 4, and$a_n=1+4n$.

Step-by-step solution

Step 1. Given Information.

The fourth term is 17 and the thirteenth term is 53.

Step 2. Write the equation for the fourth and thirteenth terms.  

Use the arithmetic sequence formula as we have $a_4=17$and $a_{13}=53$.

role="math" localid="1645285574436" $\begin{array}{rcl}{a}_4& =& a_1+\left(4-1\right)d\\ 17& =& a_1+3d\\ a_1& =& 17-3d\left(1\right)\end{array}$

And,

role="math" localid="1645285613425" $\begin{array}{rcl}{a}_{13}& =& a_1+\left(13-1\right)d\\ 53& =& a_1+12d\\ a_1& =& 53-12d\left(2\right)\end{array}$

Step 3. Find the value of common difference d.

Put equations 1 and 2 equal to find d.

$\begin{array}{rcl}17-3d& =& 53-12d\\ 12d-3d& =& 53-17\\ 9d& =& 36\\ d& =& 4\end{array}$

Step 4. Find the value of the first term.

Substitute 4 for d in equation 1.

$\begin{array}{rcl}{a}_1& =& 17-3\left(4\right)\\ & =& 17-12\\ & =& 5\end{array}$

Step 5. Find the general term.

Substitute 5 for the first term and 4 for d in the arithmetic formula.

$\begin{array}{rcl}{a}_n& =& 5+\left(n-1\right)4\\ & =& 5+4n-4\\ & =& 1+4n\end{array}$