# This sequence is geometric, what is the next term 500, 250, 125, 62.5,...?

Mar 11, 2016

The next term in the series (5th term) : $= 31.25$

#### Explanation:

$500 , 250 , 125 , 62.5$

When provided with a geometric sequence we must first calculate the common ratio $r :$

$r$ is obtained by dividing a term by its preceding term:

1) $\frac{250}{500} = \frac{1}{2}$

2) $\frac{125}{250} = \frac{1}{2}$
For the sequence the common ratio $r = \frac{1}{2}$.

Method 1:
the first term $a = 500$, and $n = 5$ (We require the 5th term)

The 5th term can be obtained through formula:

${T}_{n} = a {r}^{n - 1} = 500 \times {\left(\frac{1}{2}\right)}^{\left(5 - 1\right)}$
$= 500 \times {\left(\frac{1}{2}\right)}^{4}$

$= 500 \times \frac{1}{16} = \frac{500}{16}$

${T}_{5} = 31.25$

Method 2
We can find the fifth term by multiplying the 4th term by $r$

${T}_{5} = {T}_{4} \times \frac{1}{2}$

$= 62.5 \times \frac{1}{2}$

$= \frac{62.5}{2}$

$= 31.25$