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Try this beautiful problem from the Pre-RMO, 2017 based on Geometric Progression and integers.

## Geometric Progression and Integers – PRMO 2017

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.

- is 107
- is 48
- is 840
- cannot be determined from the given information

**Key Concepts**

Geometric series

Integers

Algebra

## Check the Answer

But try the problem first…

Answer: is 48.

Source

Suggested Reading

PRMO, 2017, Question 5

Higher Algebra by Hall and Knight

## Try with Hints

First hint

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

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

Second Hint

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

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

Final Step

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

\(\Rightarrow 5n=240\)

\(\Rightarrow n=48\).

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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