Question

The geometric mean of 1, 2, 2², ....... , 2ⁿ is

(a) \(2^{\frac{n}{2}}\)

(b) \(n^{\frac{(n+1)}{2}}\)

(c) \(2^{\frac{(n-1)}{2}}\)

(d) \(2^{\frac{n(n+1)}{2}}\)

Answer 1

(a) \(2^{\frac{n}{2}}\)

In the G.P, 1, 2, 2², ....., 2ⁿ, there are (n + 1) terms. 

∴ Geometric mean of this G.P. = (1 x 2 x 2² x ... x 2ⁿ)^{\(\frac{1}{n+1}\)}

= \((2^{1+2+3+....+n})^{\frac{1}{n+1}}\)

= \(\big(2^{\frac{n}{2}(n+1)}\big)^{\frac{1}{n+1}}\) = \(2^{\frac{n}{2}}\).