Question

Find the first term and common difference of a sequence where the third term is 2 and the twelfth term is −25. Give the formula for the general term.

Short answer

The first term, common difference, and the general formula are 8, -3, and $a_n=-3n+11$.

Step-by-step solution

Step 1. Given Information.

The third term is 2 and the twelfth term is −25.

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

Use the arithmetic sequence formula as we have $a_3=2$and $a_{12}=-25$.

role="math" localid="1645286428091" $\begin{array}{rcl}{a}_3& =& a_1+\left(3-1\right)d\\ 2& =& a_1+2d\\ a_1& =& 2-2d\left(1\right)\end{array}$

And,

role="math" localid="1645287295366" $\begin{array}{rcl}{a}_{12}& =& a_1+\left(12-1\right)d\\ -25& =& a_1+11d\\ a_1& =& -25-11d\left(2\right)\end{array}$

Step 3. Find the value of common difference d.

Put equations 1 and 2 equal to findd.

$\begin{array}{rcl}2-2d& =& -25-11d\\ 11d-2d& =& -25-2\\ 9d& =& -27\\ d& =& -3\end{array}$

Step 4. Find the value of the first term. 

Substitute -3 for d in equation 1.

$\begin{array}{rcl}{a}_1& =& 2-2\left(-3\right)\\ & =& 2+6\\ & =& 8\end{array}$

Step 5. Find the general term. 

Substitute 8 for the first term and -3 for d in the arithmetic formula.

$\begin{array}{rcl}{a}_n& =& 8+\left(n-1\right)\left(-3\right)\\ & =& 8-3n+3\\ & =& 11-3n\end{array}$