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Algebra Arithmetic Math Olympiad PRMO

Geometric Progression and Integers | PRMO 2017 | Question 5

Try this beautiful problem from the Pre-RMO, 2017 based on Geometric Progression and Integers. You may use sequential hints to solve the problem.

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


Answer: is 48.

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

\(\Rightarrow 5n=240\)

\(\Rightarrow n=48\).

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