# How do you write a rule for the nth term of the arithmetic sequence given a_8=12, a_15=61?

Aug 19, 2016

The rule for this A.P. is ${a}_{n} = 7 n - 44$

#### Explanation:

To write a rue for an AP, we need to have the value for the first term,${a}_{1} \mathmr{and} d$, the common difference.

Every term can be written in the general form of an AP,

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

In the terms we have been given we only know the values of $n$ and ${a}_{n}$

Make 2 equations:

$\text{ "a_8 = a_1 + 7d = 12} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . A$
$\text{ "a_15 = a_1 +14d = 61} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . B$

$B - A : \text{ } 7 d = 49$
$\text{ } d = 7$

Substitute 70 for d in A

${a}_{1} + 7 \left(7\right) = 12 \text{ } \Rightarrow {a}_{1} = - 37$

${a}_{n} = - 37 + \left(n - 1\right) \left(7\right)$

${a}_{n} = - 37 + 7 n - 7$

The rule for this A.P. is ${a}_{n} = 7 n - 44$