What is the geometric mean of the sequence `1,2, 4, 8,..., 2^n`

Options

A. `2^(n//2`
B. `2^((n+1)//2)`
C. `2^((n+1))-1`
D. `2^((n-1))`

Answer

A

Explanation

Geometric mean `=root(n+1)(1.2.4.8...2^(2))`
`(because" there are "(n+1)" multiples from "2^(@)" to "2^(n))`
`=root(n+1)(2^(0).2^(1).2^(2).2^(3)...2^(n))`
`=root(n+1)(2^(1+2+3+...n))=root(n+1)(2^((n(n+1))/(2)))`
`=[2^((n(n+1))/(2))]^((1)/(n+1))=2^((n)/(2))`