# How do you find the formula for a_n for the arithmetic sequence a_1=5, a_4=15?

Aug 9, 2017

${a}_{n} = \frac{10}{3} n + \frac{5}{3}$

#### Explanation:

$\text{the nth term of an arithmetic sequence is }$

•color(white)(x)a_n=a_1+(n-1)d

$\text{where d is the common difference}$

$\text{given "a_4=15" and } {a}_{1} = 5$

$\Rightarrow 5 + 3 d = 15 \Rightarrow d = \frac{10}{3}$

$\Rightarrow {a}_{n} = 5 + \frac{10}{3} \left(n - 1\right)$

$\textcolor{w h i t e}{\Rightarrow {a}_{n}} = 5 + \frac{10}{3} n - \frac{10}{3}$

$\textcolor{w h i t e}{\Rightarrow {a}_{n}} = \frac{10}{3} n + \frac{5}{3}$