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(C) If $n$ geometric means $g_1, g_2, \dots, g_n$ are inserted between two positive real numbers $a$ and $b$,then the sequence $a, g_1, g_2, \dots, g_

Vedclass · Accepted Answer

$a \left( \frac{b}{a} \right)^{\frac{n}{n+1}}$

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(C) If $n$ geometric means $g_1, g_2, \dots, g_n$ are inserted between two positive real numbers $a$ and $b$,then the sequence $a, g_1, g_2, \dots, g_n, b$ forms a $G.P.$ Let $r$ be the common ratio. The total number of terms in this $G.P.$ is $n+2$. Thus,the last term $b = a \cdot r^{(n+2)-1} = a \cdot r^{n+1}$. This implies $r^{n+1} = \frac{b}{a}$,so $r = \left( \frac{b}{a} \right)^{\frac{1}{n+1}}$. The $n^{th}$ geometric mean is $g_n = a \cdot r^n$. Substituting the value of $r$,we get $g_n = a \left( \left( \frac{b}{a} \right)^{\frac{1}{n+1}} \right)^n = a \left( \frac{b}{a} \right)^{\frac{n}{n+1}}$.

If $n$ geometric means are inserted between $a$ and $b$,then the $n^{th}$ geometric mean will be:

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