If {G₁} and {G₂} are two geometric means and A arithmetic mean inserted between two numbers, the

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Gujarati

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8. Sequences and Series

medium

If ${G_1}$ and ${G_2}$ are two geometric means and $A$ the arithmetic mean inserted between two numbers, then the value of $\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}$is

$\frac{A}{2}$

$A$

$2A$

None of these

Solution

$\therefore \,\,{G_1} = {p^{2/3}}\,\,{q^{1/3}},\,\,{G_2} = {p^{1/3}}\,\,\,{q^{2/3}}$

$\therefore \,\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}} = \frac{{{p^{4/3}}\,\,{q^{2/3}}}}{{{p^{1/3}}\,\,{q^{2/3}}}} + \frac{{{p^{2/3}}\,{q^{4/3}}}}{{{p^{2/3}}\,{q^{1/3}}}}$

$ = p + q = 2 \times \,\left( {\frac{{p + q}}{2}} \right)\, = 2A$.

Standard 11

Mathematics

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normal

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medium

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normal

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hard

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