Question

How do you find the sum of the infinite geometric series $256 + 192 + 144 + 108 + ...?$

Answer 1

Hint: The given series is an infinite geometric series and as we know that the common ratio of consecutive terms in a geometric series is fixed. To find the sum of infinite geometric series, firstly, find the common ratio between terms, by dividing a term with its progressive term.

Formula used:
1) Common ratio of a G.P.: $r = \dfrac{{{u_{n + 1}}}}{{{u_n}}}$
2) Infinite sum of G.P.: ${S_\infty } = \dfrac{a}{{1 - r}}$, where ${S_\infty }$, $a$ and $r$ are the sum of infinite geometric series, first term and the common ratio of the series respectively.

Complete step by step solution:
In order to find the sum of the infinite geometric series $256 + 192 + 144 + 108 + ...$ we will first find the common ratio of the series as following
$r = \dfrac{{{u_{n + 1}}}}{{{u_n}}},\;{\text{where}}\;{u_{n + 1}}\;{\text{and}}\;{u_n}$ are “n+1th” and “nth” term of the geometric series respectively
We will take second and first term, to find the common ratio,
$ \Rightarrow r = \dfrac{{192}}{{256}} = \dfrac{3}{4}$
Now, we will use the formula for sum of infinite terms of geometric series which is given as follows
${S_\infty } = \dfrac{a}{{1 - r}},\;{\text{where}}\;{S_\infty },\;a\;{\text{and}}\;r$ are sum of infinite geometric series, first term of the series and common ratio of the series respectively
In the series $256 + 192 + 144 + 108 + ...$ the first term is $a = 256$
Putting $a = 256\;{\text{and}}\,r = \dfrac{3}{4}$ in the above formula, we will get
${S_\infty } = \dfrac{{256}}{{1 - \dfrac{3}{4}}} = \dfrac{{256 \times 4}}{{4 - 3}} = 1024$
Therefore the required infinite sum of the given series is equals to $1024$.

Note:
Value of common ratio in a geometric should never be equals to one, because when it equals one then the progression in terms will not occur and hence geometric series will not exist. Common ratio equals to one in geometric series is equivalent to common difference equals to zero in an arithmetic series.