(A) Let the $n$ geometric means be $G_1, G_2, \dots, G_n$. Then the sequence $a, G_1, G_2, \dots, G_n, b$ forms a geometric progression with $n+2$ ter

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

(A) Let the $n$ geometric means be $G_1, G_2, \dots, G_n$. Then the sequence $a, G_1, G_2, \dots, G_n, b$ forms a geometric progression with $n+2$ terms. Let $r$ be the common ratio. The $(n+2)$-th term is given by $T_{n+2} = a \cdot r^{(n+2)-1} = a \cdot r^{n+1}$. Since $T_{n+2} = b$,we have $b = a \cdot r^{n+1}$. Therefore,$r^{n+1} = \frac{b}{a}$. Thus,$r = (\frac{b}{a})^{\frac{1}{n+1}}$.

Class 11 Mathematics Sequences and Series Geometric progression

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If there are $n$ geometric means between $a$ and $b$,what is the common ratio?

A
$(\frac{b}{a})^{\frac{1}{n+1}}$
B
$(\frac{a}{b})^{\frac{1}{n+1}}$
C
$(\frac{b}{a})^{\frac{1}{n}}$
D
$(\frac{a}{b})^{\frac{1}{n}}$

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Geometric progression

If the product of three terms of a $G.P.$ is $512$. If $8$ is added to the first term and $6$ is added to the second term,the resulting numbers are in $A.P.$. Find the numbers.

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If $\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}$ $(x \ne 0)$,then $a, b, c, d$ are in

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For a $G$.$P$.,if $S_n = \frac{4^n - 3^n}{3^n}$,then $t_2 = ........$

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If there are $n$ geometric means between $a$ and $b$,what is the common ratio?

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