kth geometric mean between two numbers for JEE

Formula

\(G_k=ar^k=a\left(\frac{b}{a}\right)^{\!\frac{k}{n+1}}\)

Variables: \(a,b\): given numbers, \(n\): number of geometric means inserted, \(k=1,2,\dots,n\), \(G_k\): kth geometric mean.

Conditions: When \(a, G_1,\dots,G_n,b\) form a GP; expression real where defined.

Finding specific inserted geometric means.