Question

How do you find the sum of the infinite geometric series $18,12,8$?

Answer 1

First we will specify all the given terms and then evaluate the values of the required terms from the question. Then we evaluate the value of common ratio and solve for the value of the sum of the series. We will be using the following formula here: ${S_n} = a\dfrac{{{r^n} - 1}}{{r - 1}}$ when $r > 1$.

Complete step-by-step solution:
Now in these kinds of questions, we start by first mentioning the given terms first. So here, we can write, $a = 50$
Here, the value of ${a_1}$ is $18$ and the value of ${a_2}$ is $12$.
Hence, the value of common ratio will be,
$\Rightarrow r = \dfrac{{{a_2}}}{{{a_1}}} \\\ \Rightarrow \dfrac{{12}}{{18}} \\\ \Rightarrow \dfrac{2}{3} \\\$
Hence, the value of $r$ is $\dfrac{2}{3}$.
Now, we mention all the evaluated terms.
$\Rightarrow a = 50 \\\ \Rightarrow r = \dfrac{2}{3} \\\$
So, now substitute all these terms in our equation of sum of all the terms.
As, here $n \to \infty$ hence, the formula becomes, $\dfrac{a}{{1 - r}}$
$\Rightarrow {S_n} = \dfrac{a}{{1 - r}} \\\ \Rightarrow {S_n} = \dfrac{{18}}{{1 - \dfrac{2}{3}}} \\\ \Rightarrow {S_n} = \dfrac{{18}}{{\dfrac{1}{3}}} \\\ \Rightarrow {S_n} = 18 \times 3 \\\ \Rightarrow {S_n} = 54 \\\$
Hence, the sum of the infinite geometric series is $54$.

Additional information: An infinite geometric is the sum of an infinite geometric sequence. This series has no last term. We can find the sum of all finite geometric series. But in the case of an infinite geometric series when the common ratio is greater than one, the terms in the sequence will get larger and larger and if you add the larger numbers, you will not get a final answer.

Note: While taking terms from one side to another, make sure you are changing their respective signs as well. While opening any brackets, always multiply the signs present outside the brackets along with the terms. Reduce the terms using the factorisation method.