If a 1 x = b 1 y = c 1 z and a , b , c are in G.P., then which of the following are true? Options: A . &nbsp x , y , z will be in A.P. B . &nbsp x , y , z will be in G.P. C . &nbsp x , y , z will be in H.P. D . &nbsp None of the above

Answer: Option A : A Let a 1 x = b 1 y = c 1 z = k . ⇒ a = k x , b = k y , c = k z . Since a , b , c are in G.P., b 2 = a c . ∴ k 2 y = k x . k z = k x + z ∴ 2 y = x + z ⇒ x , y , z are in A.P.

If a1x=b1y=c1z and a,b,c are in G.P., then which of the following are true?

Discussion Forum : Sequences And Series

Question -

If $a^{\frac{1}{x}} = b^{\frac{1}{y}} = c^{\frac{1}{z}}$ and $a, b, c$ are in G.P., then which of the following are true?

Options:

A .

x, y, z

will be in A.P.

B .

x, y, z

will be in G.P.

C .

x, y, z

will be in H.P.

D . None of the above

Answer: Option A
:
A

Let $a^{\frac{1}{x}} = b^{\frac{1}{y}} = c^{\frac{1}{z}} = k .$
$\Rightarrow a = k^{x}, b = k^{y}, c = k^{z} .$

Since $a, b, c$ are in G.P., $b^{2} = a c .$
$∴ k^{2 y} = k^{x} . k^{z} = k^{x + z}$
$∴ 2 y = x + z$
$\Rightarrow x, y, z$ are in A.P.

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Discussion Forum : Sequences And Series

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