Question

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

The second term is 13 and the tenth term is −51.

Short answer

The first term, common difference, and the general formula are 21, -8, and $a_n=-8n+29$.

Step-by-step solution

Step 1. Given Information.

The second term is 13 and the tenth term is −51.

Step 2. Write the equation for the second and tenth terms.    

Use the arithmetic sequence formula as we have $a_2=13$and $a_{10}=-51$.

$\begin{array}{rcl}{a}_2& =& a_1+\left(2-1\right)d\\ 13& =& a_1+d\\ a_1& =& 13-d\left(1\right)\end{array}$

And,

role="math" localid="1645366038048" $\begin{array}{rcl}{a}_{10}& =& a_1+\left(10-1\right)d\\ -51& =& a_1+9d\\ a_1& =& -51-9d\left(2\right)\end{array}$

Step 3. Find the value of common difference d. 

Put equations 1 and 2 equal to find d.

$\begin{array}{rcl}13-d& =& -51-9d\\ 9d-d& =& -51-13\\ 8d& =& -64\\ d& =& -8\end{array}$

Step 4. Find the value of the first term.

Substitute -8 for d in equation 1.

$\begin{array}{rcl}{a}_1& =& 13-\left(-8\right)\\ & =& 21\end{array}$

Step 5. Find the general term.

Substitute 21 for the first term and -8 for d in the arithmetic formula.

$\begin{array}{rcl}{a}_n& =& 21+\left(n-1\right)\left(-8\right)\\ & =& 21-8n+8\\ & =& -8n+29\end{array}$