# How do you write the explicit formula for the sequence 0.5,-0.1,0.02,-0.004...?

Mar 9, 2016

Explicit formula for the sequence's ${n}^{t h}$ term is $0.5 \times {\left(- \frac{1}{5}\right)}^{n - 1}$
The sequence $\left\{0.5 , - 0.1 , 0.02 , - 0.004 , . .\right\}$ is a geometric series of the type $\left\{a , a , a {r}^{2} , a {r}^{3} , \ldots .\right\}$, in which $a$ - the first term is $0.5$ and ratio $r$ between a term and its preceding term is $- \frac{1}{5}$.
As the ${n}^{t h}$ term and sum up to $n$ terms of the series $\left\{a , a , a {r}^{2} , a {r}^{3} , \ldots .\right\}$ is $a {r}^{n - 1}$ and $\frac{a \left(1 - {r}^{n}\right)}{1 - r}$ (as $r < 1$ - in case $r > 1$ one can write it as $\frac{a \left({r}^{n} - 1\right)}{r - 1}$.
As such ${n}^{t h}$ term of the given series $\left\{0.5 , - 0.1 , 0.02 , - 0.004 , . .\right\}$ is $0.5 \times {\left(- \frac{1}{5}\right)}^{n - 1}$