If frac{a}{b},frac{b}{c},frac{c}{a} are in H.P. , then

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8. Sequences and Series

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If $\frac{a}{b},\frac{b}{c},\frac{c}{a}$ are in $H.P.$, then

A

${a^2}b,\,{c^2}a,\,{b^2}c$ are in $A.P.$

B

${a^2}b,\,{b^2}c,\,{c^2}a$ are in $H.P.$

C

${a^2}b,\,{b^2}c,\,{c^2}a$ are in $G.P.$

D

None of these

Solution

(a) $\frac{b}{a},\frac{c}{b},\frac{a}{c}$ are in $A.P.$

==> $\frac{{2c}}{b} = \frac{b}{a} + \frac{a}{c}$

$ \Rightarrow \frac{{2c}}{b} = \frac{{bc + {a^2}}}{{ac}}$

==> $2a{c^2} = {b^2}c + b{a^2}$

$\therefore \,{a^2}b,\,{c^2}a$ and ${b^2}c$ are in $A.P.$

Standard 11

Mathematics

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