Question

Find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

The third term is 18 and the fourteenth term is 73.

Short answer

The first term, common difference, and the general formula are 16, 5, and$a_n=5n+11$.

Step-by-step solution

Step 1. Given Information. 

The third term is 18 and the fourteenth term is 73.

Step 2. Write the equation for the third and fourteenth terms.   

Use the arithmetic sequence formula as we have $a_3=18$and $a_{14}=73$.

role="math" localid="1645364009726" $\begin{array}{rcl}{a}_3& =& a_1+\left(3-1\right)d\\ 18& =& a_1+2d\\ a_1& =& 18-2d\left(1\right)\end{array}$

And,

$\begin{array}{rcl}{a}_{14}& =& a_1+\left(14-1\right)d\\ 73& =& a_1+13d\\ a_1& =& 73-13d\left(2\right)\end{array}$

Step 3. Find the value of common difference d. 

Put equations 1 and 2 equal to find d.

$\begin{array}{rcl}18-2d& =& 73-13d\\ 13d-2d& =& 73-18\\ 11d& =& 55\\ d& =& 5\end{array}$

Step 4. Find the value of the first term.

Substitute 5 for d in equation 1.

$\begin{array}{rcl}{a}_1& =& 18-2\left(1\right)\\ & =& 16\end{array}$

Step 5. Find the general term.

Substitute 16 for the first term and 5 for d in the arithmetic formula.

$\begin{array}{rcl}{a}_n& =& 16+\left(n-1\right)5\\ & =& 16+5n-5\\ & =& 5n+11\end{array}$