Geometric Progression and Integers | PRMO 2017 | Question 5

Let u,v,w be real numbers in geometric progression such that u>v>w. Suppose \(u^{40}=v^{n}=w^{60}\), find value of n.

Let u=a, v=ar, w=\(ar^{2}\)

then \(a^{40}\)=\((ar)^{n}\)=\((ar^{2})^{60}\)

\(\Rightarrow a^{20}=r^{-120}\)

\(\Rightarrow a=r^{-6}\)

and \(r^{-240}=r^{-5n}\)

\(\Rightarrow 5n=240\)

\(\Rightarrow n=48\).

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