If {a^{1/x}} = {b^{1/y}} = {c^{1/z}} and a,b,c are in G.P. , then x,y,z will be in

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8. Sequences and Series

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If ${a^{1/x}} = {b^{1/y}} = {c^{1/z}}$ and $a,\;b,\;c$ are in $G.P.$, then $x,\;y,\;z$ will be in

A

$A.P.$

B

$G.P.$

C

$H.P.$

D

None of these

(IIT-1969)

Solution

(a) Let ${a^{1/x}} = {b^{1/y}} = {c^{1/z}} = k$

$\Rightarrow a = {k^x},\,b = {k^y},\;c = {k^z}$

Now, $a,\;b,\;c$are in $G.P.$

$ \Rightarrow $ ${b^2} = ac $

$\Rightarrow {k^{2y}} = {k^x}.{k^z} = {k^{x + z}} $

$\Rightarrow 2y = x + z$

$ \Rightarrow $ $x,\;y,\;z$ are in $A.P.$

Standard 11

Mathematics

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