# How do you write a rule for the nth term of the arithmetic sequence and then find a_22 given d=3.8, a_8=76.4?

Jun 21, 2017

${n}^{t h}$ term is $3.8 n + 46$ and ${a}_{22} = 129.6$

#### Explanation:

In an arithmetic sequence, whose first term is ${a}_{1}$ and common difference is $d$, ${n}^{t h}$ term is given by

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

As ${a}_{8} = 76.4$, ${8}^{t h}$ term is $76.4$ or

$76.4 = {a}_{1} + \left(8 - 1\right) \times 3.8$

or $76.4 = {a}_{1} + 26.6$ and ${a}_{1} = 76.4 - 26.6 = 49.8$

and ${n}^{t h}$ term is $49.8 + \left(n - 1\right) \times 3.8 = 3.8 n + 49.8 - 3.8 = 3.8 n + 46$

hence ${a}_{22} = 3.8 \times 22 + 46 = 83.6 + 46 = 129.6$