Correct Answer - Option 1 : 3
Concept:
Geometric Progression (GP):
- The series of numbers where the ratio of any two consecutive terms is the same is called a Geometric Progression.
- A Geometric Progression of n terms with first term a and common ratio r is represented as:
a, ar, ar2, ar3, ..., arn-2, arn-1.
- The sum of the first n terms of a GP is: Sn = \(\rm a\left(\dfrac{r^n-1}{r-1}\right)\).
- The sum to ∞ of a GP, when |r| < 1, is: S∞ = \(\rm \dfrac{a}{1-r}\).
Calculation:
Let us consider the infinite series \(\rm \dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ ... \infty\).
Here, a = \(\rm \dfrac{1}{3}\) and r = \(\rm \dfrac{\tfrac{1}{9}}{\tfrac{1}{3}}=\dfrac{1}{3}\).
∴ S∞ = \(\rm \dfrac{a}{1-r}=\dfrac{\tfrac{1}{3}}{1-\tfrac{1}{3}}=\dfrac{\tfrac{1}{3}}{\tfrac{2}{3}}=\dfrac{1}{2}\).
Now, let P = \(\rm 9^{\tfrac{1}{3}} 9^{\tfrac{1}{9}} 9^{\tfrac{1}{27}}\ ...\ \infty\).
∴ P = \(\rm 9^{\tfrac{1}{3}+\tfrac{1}{9}+\tfrac{1}{27}+\ ... \infty}=9^{\tfrac{1}{2}}=\sqrt9=3\).