Let the harmonic mean and geometric mean of two positive numbers be in the ratio 4:5. Then the two numbers are in the ratio.

Let a and b be the given positive numbers then\ (H / G)=frac2 a b /(a+b)sqrt(a b)=(4 / 5) \ => 5 sqrt(a b)=2(a+b) \ => 4 a^2-17 a b+4 b^2=0 \ (a / b)=4 or (1 / 4) Therefore, the ratio is 4: 1 or 1: 4

Let the harmonic mean and geometric mean of two positiv... | Sequence & Series (Mathematics)

Let $a$ and $b$ be the given positive numbers then $\\ \frac{H}{G}=\frac{2 a b /(a+b)}{\sqrt{a b}}=\frac{4}{5} \\ \Rightarrow 5 \sqrt{a b}=2(a+b) \\ \Rightarrow 4 a^2-17 a b+4 b^2=0$ $\\ \frac{a}{b}=4 \text { or } \frac{1}{4}\\$ Therefore, the ratio is $4: 1$ or $1: 4$

Explanation

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