Continued fraction

It is not that difficult
Ok here is the solution:

If we start solving the continued fraction from below we will find that-

1+1x=x+1x

1+1x+1x=1+xx+1=2x+1x+1

1+12x+1x+1=1+x+12x+1=3x+22x+1

After solving a few more of those we will find the other freactions are 5x+33x+2,8x+55x+3 and hence the
fibonnaci series is becoming visible as each of the fractions are of the form :

f_n+1x + f_nf_nx + f_n-1 where f_n is the nth term of the fibonacci series

This is because:

1+1f_n+1x + f_nf_nx + f_n-1 =f_n+2x + f_n+1f_n+1x + f_n

thus we find that

f_n+1x + f_nf_nx + f_n-1 =x

on solving which we find the quadratic equation:

x²-x-1=0 hence x =1+/-√52

Another way of solving is if we assume that x+1x=x

The LHS does converge because:

f_n+1x + f_nf_nx + f_n-1 =2f_nx + f_n-1f_nx + f_n-1 =f_nxf_nx + f_n-1 +1

Since f_nxf_nx + f_n-1 converges the LHS converges

I guess its rather 1+\frac{1}{1+x}=x , which gives x^2=2 .

you know, before you do that neat trick, you have the responsibility to prove that that expression on LHS actually converges.

Sir I never gave much thought seeing it under the IX-X section.
@ Prophet Sir - I have a very vague idea abt convergence. Can you please tell me where I can read about it, somewhat easy explanation?

@Sri & @Ashish the question simplifies to 1+1x=x

This was the easy proof...can you give me another proof.(hint: start solving the continued fraction and you will find a form of a fibonnaci series)