(A) Let $a^{1/x} = b^{1/y} = c^{1/z} = k$ (where $k \neq 1$).Then $a = k^x, b = k^y, c = k^z$.Since $a, b, c$ are in geometric progression,we have $b^

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(A) Let $a^{1/x} = b^{1/y} = c^{1/z} = k$ (where $k \neq 1$).Then $a = k^x, b = k^y, c = k^z$.Since $a, b, c$ are in geometric progression,we have $b^2 = ac$.Substituting the values,we get $(k^y)^2 = k^x \cdot k^z$.$k^{2y} = k^{x+z}$.Equating the exponents,$2y = x + z$.This implies that $x, y, z$ are in arithmetic progression.

Class 11 Mathematics Sequences and Series Relation between A.P., G.P.

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If $a^{1/x} = b^{1/y} = c^{1/z}$ and $a, b, c$ are in geometric progression,then $x, y, z$ are in.....

A
Arithmetic progression.
B
Geometric progression.
C
Harmonic progression.
D
None of these.

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Relation between A.P., G.P.

If the $2^{nd}, 5^{th},$ and $9^{th}$ terms of a non-constant $A.P.$ are in $G.P.$,then the common ratio of this $G.P.$ is:

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If $2(y - a)$ is the $H.M.$ between $y - x$ and $y - z$,then $x - a, y - a, z - a$ are in

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If $a_1, a_2, \dots, a_n$ are positive real numbers such that $a_1 \cdot a_2 \cdot \dots \cdot a_n = 1$,then their sum is:

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If $\alpha, \beta, \gamma$ are the geometric means between $ca, ab$; $ab, bc$; and $bc, ca$ respectively,where $a, b, c$ are in $A.P.$,then $\alpha^2, \beta^2, \gamma^2$ are in

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If $a, b, c$ are in $G.P.$ and $a^{\frac{1}{x}} = b^{\frac{1}{y}} = c^{\frac{1}{z}} = k,$ prove that $x, y, z$ are in $A.P.$

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If $a^{1/x} = b^{1/y} = c^{1/z}$ and $a, b, c$ are in geometric progression,then $x, y, z$ are in.....

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