Question

The sum of the infinite series \[1 + 2 + \dfrac{2}{3} + \dfrac{6}{{{3^2}}} + \dfrac{{10}}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} + .......\] is:
A) 6
B) \[\dfrac{9}{2}\]
C) 4
D) 5

Answer 1

Hint:
Here we will first assume the fractional part of the infinite series as a variable. Then we will find the value of that variable by converting it in terms of the sum of the G.P. series. We will use the basic formula of the sum of the G.P. series to get the value of that variable. Then by using the value of that variable we will get the value of the sum of the given infinite series.

Formula used:
We will use the formula of sum of an infinite G.P. series \[{S_\infty } = \dfrac{a}{{1 - r}}\], where \[a\] is the first term and \[r\] is the common ratio.

Complete Step by Step Solution:
Given series is \[1 + 2 + \dfrac{2}{3} + \dfrac{6}{{{3^2}}} + \dfrac{{10}}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} + .......\]
Let \[S = 1 + 2 + \dfrac{2}{3} + \dfrac{6}{{{3^2}}} + \dfrac{{10}}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} + .......\]
Adding the first two terms, we get
\[ \Rightarrow S = 3 + \dfrac{2}{3} + \dfrac{6}{{{3^2}}} + \dfrac{{10}}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} + .......\]
We can write the above equation as
\[ \Rightarrow S = 3 + L\]……………………….\[\left( 1 \right)\]
Here, \[L = \dfrac{2}{3} + \dfrac{6}{{{3^2}}} + \dfrac{{10}}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} + .......\] …………………….\[\left( 2 \right)\]

Now we will take the expression of \[L\] and find its value.
We will divide the expression of \[L\] by 3. Therefore, we get
\[\dfrac{L}{3} = \dfrac{2}{{{3^2}}} + \dfrac{6}{{{3^3}}} + \dfrac{{10}}{{{3^4}}} + \dfrac{{14}}{{{3^5}}} + .......\]……………………….\[\left( 3 \right)\]
Now we will subtract the equation \[\left( 3 \right)\] from equation \[\left( 2 \right)\]. Therefore, we get
\[ \Rightarrow L - \dfrac{L}{3} = \left( {\dfrac{2}{3} + \dfrac{6}{{{3^2}}} + \dfrac{{10}}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} + .......} \right) - \left( {\dfrac{2}{{{3^2}}} + \dfrac{6}{{{3^3}}} + \dfrac{{10}}{{{3^4}}} + \dfrac{{14}}{{{3^5}}} + .......} \right)\]
\[ \Rightarrow \dfrac{{2L}}{3} = \left( {\dfrac{2}{3} + \dfrac{6}{{{3^2}}} - \dfrac{2}{{{3^2}}} + \dfrac{{10}}{{{3^3}}} - \dfrac{6}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} - \dfrac{{10}}{{{3^4}}} + .......} \right)\]
Simplifying the above equation, we get
\[ \Rightarrow \dfrac{{2L}}{3} = \left( {\dfrac{2}{3} + \dfrac{4}{{{3^2}}} + \dfrac{4}{{{3^3}}} + \dfrac{4}{{{3^4}}} + .......} \right)\]
\[ \Rightarrow \dfrac{{2L}}{3} = \dfrac{2}{3} + 4\left( {\dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} + \dfrac{1}{{{3^4}}} + .......} \right)\]
We can clearly see that the term \[\left( {\dfrac{1}{{{3^2}}} + \dfrac{1}{{{3^3}}} + \dfrac{1}{{{3^4}}} + .......} \right)\] is a sum of a G.P. series with first term \[a = \dfrac{1}{{{3^2}}}\] and common ratio \[r = \dfrac{1}{3}\]. so, we will use the formula of the sum of the G.P. series to get its value.
Using the formula, \[{S_\infty } = \dfrac{a}{{1 - r}}\], we get
\[ \Rightarrow \dfrac{{2L}}{3} = \dfrac{2}{3} + 4\left( {\dfrac{{\dfrac{1}{{{3^2}}}}}{{1 - \dfrac{1}{3}}}} \right)\]
Subtracting the terms in the denominator, we get
\[ \Rightarrow \dfrac{{2L}}{3} = \dfrac{2}{3} + 4\left( {\dfrac{{\dfrac{1}{{{3^2}}}}}{{\dfrac{2}{3}}}} \right)\]
\[ \Rightarrow \dfrac{{2L}}{3} = \dfrac{2}{3} + 4\left( {\dfrac{1}{6}} \right)\]
Now we will solve this equation to get the value of \[L\]. Therefore, we get
\[ \Rightarrow L = \dfrac{3}{2}\left( {\dfrac{2}{3} + \dfrac{2}{3}} \right)\]
\[ \Rightarrow L = \dfrac{3}{2}\left( {\dfrac{4}{3}} \right) = 2\]
Now we will put the value of \[L\] in the equation \[\left( 1 \right)\] to get the value of the sum. Therefore, we get
\[ \Rightarrow S = 3 + 2 = 5\]

Hence, the sum of the infinite series \[1 + 2 + \dfrac{2}{3} + \dfrac{6}{{{3^2}}} +
\dfrac{{10}}{{{3^3}}} + \dfrac{{14}}{{{3^4}}} + .......\] is 5.

Note:
Here we expressed the given series in GP or geometric progression. Geometric progression is defined as the sequence or the series in which the consecutive terms differ by a common ratio. Other than geometric progression, we also have different progression such as arithmetic progression and harmonic progression. Harmonic progression is defined as the sequence or the series in which the consecutive terms have a common difference. Harmonic progression is the reciprocal of arithmetic progression.