# The third term in an arithmetic sequence is 10 and the fifth term is 26. If the first term is a1, which is an equation for the nth term of this sequence?

Oct 17, 2017

${a}_{n} = 8 n - 14$

#### Explanation:

for an AP

where

${a}_{1} =$the first term

$d =$the common difference
then we have

${a}_{1} , \left({a}_{1} + d\right) , \left({a}_{1} + 2 d\right) , \ldots , {a}_{1} + \left(\left(n - 1\right) d\right) , . .$

we are given the third term

$10 = {a}_{1} + 2 d - - \left(1\right)$

and the fifth term

$26 = {a}_{1} + 4 d - - \left(2\right)$

subtract

$\left(2\right) - \left(1\right)$

$16 = 2 d \implies d = 8$

sub into$\left(1\right)$

$10 = {a}_{1} + 2 \times 8$

$\implies {a}_{1} = - 6$

so the nth term

${a}_{n} = a + \left(n - 1\right)$

will be

${a}_{n} = - 6 + 8 \left(n - 1\right)$

${a}_{n} = 8 n - 14$