# How do you find the 10th term in the geometric sequence 25, 5, 1, 0.2, ...?

Nov 14, 2015

Write an exponential equation.

#### Explanation:

Each term of the geometric sequence is $\frac{1}{5}$ the previous term. Therefore, you can say that each term is the previous term multiplied by ${5}^{-} 1$. We can write an equation that models this, starting at $25$, where $x$ is the number term in the sequence.

$f \left(x\right) = 25 {\left(5\right)}^{-} \left(x - 1\right)$

With this equation, the first term would be calculated so: $f \left(x\right) = 25 {\left(5\right)}^{-} \left(1 - 1\right) = 25 {\left(5\right)}^{-} 0 = 25 \cdot 1 = 25$

Finding the tenth term: $f \left(x\right) = 25 {\left(5\right)}^{-} \left(10 - 1\right) = 25 {\left(5\right)}^{-} 9 = \frac{25}{{5}^{9}} = \frac{{5}^{2}}{{5}^{9}} = \textcolor{b l u e}{\frac{1}{{5}^{7}}}$.

This is equal to $\textcolor{red}{0.0000128}$.