# How do you write the formula for the nth term given 900, 300, 100, 33 1/3,...?

Apr 2, 2016

$900 \cdot {\left(\frac{1}{3}\right)}^{n - 1}$

#### Explanation:

Each term after the first is equal to the previous term multiplied by $\frac{1}{3}$. If we do not simplify at each step, and simply group the $\frac{1}{3}$s together, then we can rewrite the sequence as

$900 , 900 \cdot \frac{1}{3} , 900 \cdot {\left(\frac{1}{3}\right)}^{2} , 900 \cdot {\left(\frac{1}{3}\right)}^{3} , \ldots$

and from this it is evident that the ${n}^{\text{th}}$ term may be written as $900 \cdot {\left(\frac{1}{3}\right)}^{n - 1}$.

This type of sequence is called a geometric sequence. A geometric sequence is a sequence of the form ${a}_{0} , {a}_{0} r , {a}_{0} {r}^{2} , \ldots$ where ${a}_{0}$ is the starting term and $r$ is the common ratio between terms. In this case, we'd have ${a}_{0} = 900$ and $r = \frac{1}{3}$.