Question

if a/b=b/c, prove that: a2b2c2(1/a3+1/b3+1/c3)=a3+b3+c3​

Answer 1

ANSWER:

Given:

• a/b = b/c

To Prove:

• a²b²c²(1/a³ + 1/b³ + 1/c³) = a³ + b³ + c³

Proof:

$\text{We are given that,}\\\\:\longrightarrow\dfrac{a}{b}=\dfrac{b}{c}\\\\\text{So,}\\\\:\implies b^2=ac- - - -(1)\\\\\text{We need to prove that,}\\\\:\longrightarrow a^2b^2c^2\left(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}\right)=a^3+b^3+c^3\\\\\text{Taking LHS,}\\\\:\implies a^2b^2c^2\left(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}\right)\\\\\text{From (1),}\\\\:\implies a^2(ac)c^2\left(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}\right)\\\\:\implies a^3c^3\left(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}\right)$

$\text{Taking LCM of a^3, b^3 , c^3 :}\\\\:\implies a^3\!\!\!\!\!/\:c^3\!\!\!\!\!/\:\left(\dfrac{b^3c^3+a^3c^3+a^3b^3}{a^3\!\!\!\!\!/\:b^3c^3\!\!\!\!\!/\:}\right)\\\\:\implies\dfrac{b^3c^3+(ac)^3+a^3b^3}{b^3}\\\\\text{Using (1),}\\\\:\implies\dfrac{b^3c^3+(b^2)^3+a^3b^3}{b^3}\\\\:\implies\dfrac{b^3c^3+b^6+a^3b^3}{b^3}\\\\\text{Taking b^3 common,}\\\\:\implies\dfrac{b^3\!\!\!\!\!/\:(c^3+b^3+a^3)}{b^3\!\!\!\!\!/\:}\\\\\bf{:\implies a^3+b^3+c^3 =RHS}\\\\\text{\bf{HENCE PROVED!!!}}$