81^1/3 × 81^1/9 × 81^1/27 × 81^1/81 × ______ up to infinite term will be equal to

Concept:

Sum of a G.P with first term a and common ratio "r".

is \(\rm\frac{a}{1-r}\).

Calculation:

Given 811/3 × 811/9 × 811/27 × 811/81 × ----

⇒ 81^{1/3 + 1/9 + 1/27 + 1/81 + .....}

⇒ \(81^{\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+....}\)

Sum of a G.P with first term a and common ratio "r".

= \(\rm\frac{a}{1-r}\)

∴ we can write the equation \(81^{\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+....}\) as:

⇒ \(81^{\frac{\frac{1}{3}}{1-\frac{1}{3}}}\)

⇒ \(81^{\frac{\frac{1}{3}}{\frac{2}{3}}}\) 

= 9.

Required value of the expression is 9.