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

English

Gujarati

Hindi

8. Sequences and Series

medium

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

Share

Similar Questions

Suppose that all the terms of an arithmetic progression ($A.P.$) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is $6: 11$ and the seventh term lies in between $130$ and $140$ , then the common difference of this $A.P.$ is

hard

(IIT-2015)

Find the sum of integers from $1$ to $100$ that are divisible by $2$ or $5.$

hard

If ${a^2},\;{b^2},\;{c^2}$ are in $A.P.$, then ${(b + c)^{ – 1}},\;{(c + a)^{ – 1}}$ and ${(a + b)^{ – 1}}$ will be in

hard

If the sum of $n$ terms of an $A.P.$ is $\left(p n+q n^{2}\right),$ where $p$ and $q$ are constants, find the common difference.

medium

Given that $n$ A.M.'s are inserted between two sets of numbers $a,\;2b$and $2a,\;b$, where $a,\;b \in R$. Suppose further that ${m^{th}}$ mean between these sets of numbers is same, then the ratio $a:b$ equals

medium