If $G_{1}, G_{2}, G_{3} . . . . G_{n}$ are n numbers inserted between the numbers a and b of a G.P. then $G_{1} \times G_{2} \times G_{3} \times . . . . G_{n} =$

\sqrt{a b}

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{(\sqrt{a b})}^{n}

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{(\sqrt{a b})}^{n + 1}

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(\frac{a + b}{2})^{n}

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Solution

The correct option is B ${(\sqrt{a b})}^{n}$
let $a, G_{1}, G_{2}, G_{3}, . . . G_{n}, b$ be in G.P
$G_{1} = a r G_{2} = a r^{2} . . . . G_{n} = a r^{n}$
$G_{1} . G_{2} . . . . . . . . G_{n} = (a^{n}) . (r^{\frac{n (n + 1)}{2}}) \to [1]$
and $a \times b = (a^{2}) . (r^{(n + 1)})$
finding $r$ from second equation in terms of $a b$ and substituting it in first equation gives
$G_{1} . G_{2} . . . . . . . . G_{n} = \sqrt{{(a b)}^{n}}$

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