Question

Find the eleventh term of a sequence where the third term is 19 and the common difference is five. Give the formula for the general term.

Short answer

The eleventh term is 59 and the general term is$a_n=5n-4$.

Step-by-step solution

Step 1. Given Information. 

The third term is 19 and the common difference is five.

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

Substitute 3 for n, 5 for d, and the third term is 19, in the formula $a_n=a_1+\left(n-1\right)d$.

$\begin{array}{rcl}{a}_3& =& a_1+\left(3-1\right)5\\ 19& =& a_1+10\\ a_1& =& 9\end{array}$

Step 3. Find the eleventh term. 

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

$\begin{array}{rcl}{a}_{11}& =& 9+\left(11-1\right)5\\ & =& 9+50\\ & =& 59\end{array}$

Step 4. Find the general terms.

To find the general terms, substitute the first term 9 and the common difference 5 in the formula.

$\begin{array}{rcl}{a}_n& =& 9+\left(n-1\right)5\\ & =& 9+5n-5\\ & =& 5n-4\end{array}$