A problem from NCERT textbook

Easiest method, Just observe $n=0$ is a solution. :P

Here is the complete solution. Given,

$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\frac{a+b}{2}$ $\frac{a^{n+1}+b^{n+1}}{a^{n+1}+b^{n+a}+ab^n+ba^n}=\frac{1}{2}$

Just take its reverse and see that

$2=\frac{a^{n+1}+b^{n+a}+ab^n+ba^n}{a^{n+1}+b^{n+1}}$

$2=1+\frac{ab^n+ba^n}{a^{n+1}+b^{n+a}}$ $1=\frac{ab^n+ba^n}{a^{n+1}+b^{n+a}}$

Now, we feel no harm in cross multiplying, so, lets do it

$a^{n+1}+b^{n+1}=ab^n+ba^n$ $a^n(a-b)-b^n(a-b)=0$ So, either $a^n=b^n$ or $a=b$

But Given that $a \neq b$ , Thus $a^n=b^n$ Now taking log on both sides, we get $n=0$ is the solution..

Personally, I think my First observation is much simpler :D

A question from NCERT Class XI Mathematics Textbook!

solving the eqn. we get,a^n=B^n =>n=0