# How do you write a rule for the nth term of the arithmetic sequence given a_10=8, a_16=32?

Aug 5, 2016

$\left\{\begin{matrix}{a}_{0} = - 32 \\ {a}_{i + 1} = {a}_{i} + 4\end{matrix}\right.$

#### Explanation:

For an arithmetic sequence $< {a}_{0} , {a}_{1} , {a}_{2} , {a}_{3} , \ldots >$
terms are related by the formula:
$\textcolor{w h i t e}{\text{XXX}} {a}_{m} = {a}_{n} + \left(m - n\right) \cdot k$ for some constant $k$

In this example:
$\textcolor{w h i t e}{\text{XXX}} {a}_{16} = {a}_{10} + \left(16 - 10\right) \cdot k$
or (using the given values)
$\textcolor{w h i t e}{\text{XXX}} 32 = 8 + 6 \cdot k$

$\textcolor{w h i t e}{\text{XX}} \rightarrow k = 4$

And the initial value, ${a}_{0}$ is
$\textcolor{w h i t e}{\text{XXX}} {a}_{0} = {a}_{10} + \left(0 - 10\right) \cdot k$
or
$\textcolor{w h i t e}{\text{XXX}} {a}_{0} = 8 + \left(- 10\right) \cdot \left(4\right) = - 32$

If your standard is to use ${a}_{1}$ as an initial value:
$\textcolor{w h i t e}{\text{XXX}} {a}_{1} = {a}_{10} + \left(1 - 10\right) \cdot k$
$\textcolor{w h i t e}{\text{XXXXX}} = 8 + \left(- 9\right) \cdot \left(4\right) = - 28$