Geometric Series Examples
Here are the all important examples on Geometric Series.
What is geometric series ?
Geometric series is a series in which ratio of two successive terms is always constant.
Lets take a example.
Example : 1/2,1/4,1/8,1/16,....
Here the ratio of any two terms is 1/2 , and the series terms values get increased by factor of 1/2.
Example : 2,4,8,16,32,64..... is also an example of geometric series.
or in a general way geometric series can represented as $a,ar,ar^{2},ar^{3},ar^{4}.......$
Sum of geometric series
The formula for sum of n terms of geometric progression is
$ S_{n}=\frac{a\left(1-r^{n}\right)}{1-r} $
Formula for sum of infinite terms of geometric series
Geometric series has sum if and only if $ |r|<1 $ and this case sum is,
$ S=\frac{a}{1-r} $
Then the series is called convergent.
Examples
Example #1
Find the 8th and nth terms of the G.P. 5,25,125, ...
Here $ a=5 $ and $ r=5 . $ Thus, $ a_{8}=5(5)^{8-1}=5(5)^{7}=5^{8} $
and $ \quad a_{n}=a r^{n-1}=5(5)^{n-1}=5^{n} $
Example #2
Find the sum of first n terms and the sum of first 5 terms of the geometricseries $ 1+\frac{2}{3}+\frac{4}{9}+\ldots $
Here $ a=1 $ and $ r=\frac{2}{3} . $ Therefore
$ S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$$=\frac{\left[1-\left(\frac{2}{3}\right)^{n}\right]}{1-\frac{2}{3}}=3\left[1-\left(\frac{2}{3}\right)^{n}\right] $
In particular, $ \mathrm{S}_{5}=3\left[1-\left(\frac{2}{3}\right)^{5}\right]$$=3 \times \frac{211}{243}=\frac{211}{81} $
Example #3
Find the sum of the sequence 7,77,777,7777, ... to n terms.
This is not a G.P., however, we can relate it to a G.P. by writing the terms as
$ \mathrm{S}_{n}=7+77+777+7777+\ldots $ to $ n $ terms
$ =\frac{7}{9}[9+99+999+9999+\ldots . $ to $ n $ term $ ] $
$ =\frac{7}{9}[(10-1)+\left(10^{2}-1\right)+\left(10^{3}-1\right)+\left(10^{4}-1\right)+\ldots n $ terms]
$ =\frac{7}{9}\left[\left(10+10^{2}+10^{3}+\right.\right.$ + ... n $ terms $ )-(1+1+1+ ... n $ terms $ ) ] $ $
$ =\frac{7}{9}\left[\frac{10\left(10^{n}-1\right)}{10-1}-n\right]$$=\frac{7}{9}\left[\frac{10\left(10^{n}-1\right)}{9}-n\right] $