The value of \(\rm 64^{\tfrac{1}{3}}\cdot 64^{\tfrac{1}{9}}\cdot 64^{\tfrac{1}{27}}\ ...\ \infty\) is:

Question

Question

Answer 1

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 = .
The sum ∞ of a GP, when |r| < 1, is: S∞ = .

Calculation:

Given: \(\rm 64^{\tfrac{1}{3}}\cdot 64^{\tfrac{1}{9}}\cdot 64^{\tfrac{1}{27}}\ ...\ \infty\)

Let us consider the infinite series .

Here, a = and r = .

∴ S∞ = .

Now, let P =

∴ P = .