# How do you find a formula for the nth term of the geometric sequence: 500, 100, 20, 4, ...?

Apr 29, 2016

${t}_{n} = a {\left(\frac{1}{5}\right)}^{n - 1}$

#### Explanation:

Recall that the formula for the ${n}^{\text{th}}$ of a geometric sequence is:

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {t}_{n} = a {r}^{n - 1} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

where:
${t}_{n} =$term number
$a =$first term
$r =$common ratio
$n =$number of terms

Start by determining the value of $a$, the first term of the sequence. In your case, that would be $500$.

Use $a = 500$, and ${t}_{2} = 100$ to create an equation representing the second term.

${t}_{n} = a {r}^{n - 1}$

$100 = 500 {r}^{2 - 1}$

Solve for $r$.

$r = \frac{1}{5}$

Now that you have the value of $r$, you can make a formula representing the ${n}^{\text{th}}$ term of the geometric sequence.

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {t}_{n} = a {\left(\frac{1}{5}\right)}^{n - 1} \textcolor{w h i t e}{\frac{a}{a}} |}}}$