Question

The value of \[{{4}^{{}^{1}/{}_{3}}}\cdot {{4}^{{}^{1}/{}_{9}}}\cdot {{4}^{{}^{1}/{}_{27}}}\cdot ....\infty \] is
A. $2$
B. $3$
C. $4$
D. $9$

Answer 1

Hint: In this question, we are to find the sum of infinite terms of the given series obtained by simplifying the given sequence. By using that we can able to find the given expression.

Formula Used:The sum of the infinite terms in the G.P series is calculated by
 ${{S}_{\infty }}=\dfrac{a}{1-r}$ where $r=\dfrac{{{a}_{n}}}{{{a}_{n-1}}}$
Here ${{S}_{\infty }}$ is the sum of the infinite terms of the series; $a$ is the first term in the series, and $r$ is the common ratio.

Complete step by step solution:The given series is
\[{{4}^{{}^{1}/{}_{3}}}\cdot {{4}^{{}^{1}/{}_{9}}}\cdot {{4}^{{}^{1}/{}_{27}}}\cdot ....\infty \]
Rewriting the given series using the exponential formula \[{{a}^{m}}\cdot {{a}^{n}}={{a}^{m+n}}\],
\[{{4}^{{}^{1}/{}_{3}}}\cdot {{4}^{{}^{1}/{}_{9}}}\cdot {{4}^{{}^{1}/{}_{27}}}\cdot ....\infty ={{4}^{\left( {}^{1}/{}_{3}+{}^{1}/{}_{9}+{}^{1}/{}_{27}+...\infty \right)}}\]
Thus, a geometric series is formed in the exponential. I.e.,
\[{}^{1}/{}_{3}+{}^{1}/{}_{9}+{}^{1}/{}_{27}+...\infty \]
Here the first term \[a=\dfrac{1}{3}\];
And the common ratio \[r=\dfrac{\dfrac{1}{9}}{\dfrac{1}{3}}=\dfrac{1}{3}\]
So, the sum of infinite terms in the series is
\[\begin{align}
  & {{S}_{\infty }}=\dfrac{a}{1-r} \\
 & \text{ }=\dfrac{{}^{1}/{}_{3}}{1-{}^{1}/{}_{3}} \\
 & \text{ }=\dfrac{\dfrac{1}{3}}{\dfrac{2}{3}} \\
 & \text{ }=\dfrac{1}{2} \\
\end{align}\]
Then, the given expression is,
\[\begin{align}
  & {{4}^{{}^{1}/{}_{3}}}\cdot {{4}^{{}^{1}/{}_{9}}}\cdot {{4}^{{}^{1}/{}_{27}}}\cdot ....\infty ={{4}^{\left( {}^{1}/{}_{3}+{}^{1}/{}_{9}+{}^{1}/{}_{27}+...\infty \right)}} \\
 & \text{ }={{4}^{\left( \dfrac{1}{2} \right)}} \\
 & \text{ }=\sqrt{4} \\
 & \text{ }=2 \\
\end{align}\]


Option ‘A’ is correct

Note: Here the given series is geometric series. So, by using the appropriate formula, the sum of infinite terms is calculated. The given expression is evaluated by substituting the obtained values.