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Q 102

Expert-verified
Found in: Page 1184

### Intermediate Algebra

Book edition OER 2017
Author(s) OPENSTAX
Pages 1346 pages
ISBN 9780998625720

# 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.

The first term, common difference, and the general formula are 16, 5, and ${a}_{n}=5n+11$.

See the 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.

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