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Q 102

Expert-verifiedFound in: Page 1184

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 ${a}_{n}=5n+11$.

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

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,

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

Put equations 1 and 2 equal to find *d.*

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

Substitute 5 for *d* in equation 1.

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

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

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

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