NIMCET 2013 — Mathematics PYQ

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Question

NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

The value of $9^{\frac{1}{3}} \cdot 9^{\frac{1}{9}} \cdot 9^{\frac{1}{27}} \dots \infty$ is:

Choose the correct answer:

Explanation

Solution

1. Use the Property of Exponents:

Since the bases are the same, the powers are added:

P = 9^{\left( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty \right)}

2. Identify the Infinite Geometric Progression (G.P.) in the Power:

Let $S = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty$

First term $(a) = \frac{1}{3}$
Common ratio $(r) = \frac{1/9}{1/3} = \frac{1}{3}$

3. Calculate the Sum of the Infinite G.P. ( $S_\infty$ ):

Using the formula $S_\infty = \frac{a}{1 - r}$ (where $|r| < 1$ ):

S = \frac{\frac{1}{3}}{1 - \frac{1}{3}}

S = \frac{\frac{1}{3}}{\frac{2}{3}}

S = \frac{1}{2}

4. Substitute the Sum back into the Power:

P = 9^S

P = 9^{\frac{1}{2}}

P = \sqrt{9}

P = 3