Finding the Sum of an Infinite Geometric Sequence

An infinite geometric sequence is one in which there is no definitive last term as the sequence continues indefinitely.

The Sum to Infinity

The sum to infinity of a geometric sequence is:

$S_{\infty}=\dfrac {a}{1-r}$ where $-1<r<1$

We can use this equation to express infinite numbers as fractions.

Example 1

Express $1. \overline 2$ as a fraction.

To do this, we need to express $1. \overline 2$ as the sum of a geometric sequence.

$1. \overline 2 =1.2222222...$
$1. \overline 2 = 1+(0.2+0.02+0.002+0.0002 + ...)$

An infinite geometric sequence is formed by the terms in the bracket. So:

$a=0.2$
$r=\frac{0.02}{0.2}=0.1$

Now we can use the formula.

$S_{\infty}=\frac {a}{1-r}$
$0. \overline 2 =\frac{0.2}{1-0.1}$
$0. \overline 2 =\frac {0.2}{0.9}$
$0. \overline 2 =\frac 29$

Now all we need to do is add 1.

$1. \overline 2 =1+ \frac 29$
$1. \overline 2 = 1\frac 29$

For decreasing or decaying geometric series, the sum of an infinite number of terms will approach a finite sum.

Example 2

A tired dog is trying to find its way home. It travels 900 metres in the first hour, 720 metres in the second hour and 576 metres in the third hour. If the dog lives 4 kilometres away, will he make it home?

We first must determine what sequence we have as this information has not been provided in the question.

$t_1=900, t_2=720$ and $t_3=576$
$\frac {720}{900}=0.8$
$\frac {576}{720}=0.8$

So we know we have a geometric sequence as we have a common ratio. Now we can use the formula.

$a=900$
$r=0.8$
$S_{\infty}=\frac {900}{1-0.8}$
$S_{\infty}=4,500$

4,500m = 4.5km

The dog can travel a distance of 4.5 kilometres so he will get to his home as it is only 4 kilometres away.

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