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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Sequences and Series

Olympiad workbook · IOQM · Sequences and Series

Let u, v, w be real numbers in geometric progression such that...

Let

\mathrm{u}, \mathrm{v}, \mathrm{w}

be real numbers in geometric progression such that

\mathrm{u}>\mathrm{v}>\mathrm{w}

. Suppose

\mathrm{u}^{40}=\mathrm{v}^{\mathrm{n}}=\mathrm{w}^{60}

. Find the value of

n

.

Correct answer

48

Step-by-step solution

\begin{aligned} & u=a \\ & v=a r \\ & w=a r^2 \\ & a^{40}=(a r)^n=\left(a r^2\right)^{60} \\ & a^{20}=r^{-120} \Rightarrow a=r^{-6} \\ & r^{-240}=r^{-5 n} \Rightarrow \quad 5 n=240 \\ & n=\frac{240}{5}=48\end{aligned}

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