Try this beautiful problem from the Pre-RMO, 2017 based on Geometric Progression and integers.
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.
Geometric series
Integers
Algebra
Answer: is 48.
PRMO, 2017, Question 5
Higher Algebra by Hall and Knight
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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