Question

if abc=1 then proved that {1/(1+a+1/b)}+{1/(1+b+1/c)}+{1/(1+c+1/a)}=1​

Answer 1

ANSWER:

Given:

• abc = 1

To Prove:

$\sf\:\:\:\bullet\:\:\:\dfrac{1}{1+a+\frac{1}{b}}+\dfrac{1}{1+b\frac{1}{c}}+\dfrac{1}{1+c+\frac{1}{a}}=1$

Proof:

We need to prove that,

$\sf\implies\dfrac{1}{1+a+\frac{1}{b}}+\dfrac{1}{1+b+\frac{1}{c}}+\dfrac{1}{1+c+\frac{1}{a}}=1$

Solving LHS,

$\sf\implies\dfrac{1}{1+a+\frac{1}{b}}+\dfrac{1}{1+b+\frac{1}{c}}+\dfrac{1}{1+c+\frac{1}{a}}$

Simplifying it,

$\sf\implies\dfrac{1}{\frac{b+ab+1}{b}}+\dfrac{1}{\frac{c+bc+1}{c}}+\dfrac{1}{\frac{a+ca+1}{a}}$

$\sf\implies\dfrac{b}{b+ab+1}+\dfrac{c}{c+bc+1}+\dfrac{a}{a+ca+1}$

Now, we are given that,

⇒ abc = 1

So,

⇒ a = 1/bc

Hence,

$\sf\implies\dfrac{b}{b+ab+1}+\dfrac{c}{c+bc+1}+\dfrac{a}{a+ca+1}$

$\sf\implies\dfrac{b}{b+\left(\frac{1}{bc}\right)b+1}+\dfrac{c}{c+bc+1}+\dfrac{\frac{1}{bc}}{\frac{1}{bc}+c\left(\frac{1}{bc}\right)+1}$

$\sf\implies\dfrac{b}{b+\frac{1}{c}+1}+\dfrac{c}{c+bc+1}+\dfrac{\frac{1}{bc}}{\frac{1}{bc}+\frac{c}{bc}+1}$

$\sf\implies\dfrac{b}{\frac{bc+1+c}{c}}+\dfrac{c}{c+bc+1}+\dfrac{\frac{1}{bc}}{\frac{1+c+bc}{bc}}$

$\sf\implies\dfrac{bc}{bc+c+1}+\dfrac{c}{bc+c+1}+\dfrac{1}{1+c+bc}$

So,

$\sf\implies\dfrac{bc+c+1}{bc+c+1}$

Hence,

$\implies\bf1 = RHS$

As, LHS = RHS

HENCE PROVED!!!!!