a2 = f(a 1)= f(x/√( 1+x^2))

=x/√( 1+x^2)/√( 1+x^2/( 1+x^2))

=x/√( 1+x^2)/√(( 1+2x^2)/( 1+x^2))

=x/ √( 1+2x^2)

Similarly, A3 = x/√ (1+3x 2)

a4=x/ √( 1+4x^2)

It is easy to guess an = x/√ (1+NX 2) from the above.

Prove:

A1= x/√ (1+x 2) = x/√ (1+1x 2), which holds.

Assuming that x=n holds, then an = x/√ (1+NX 2),

When x=n+ 1,

a(n+ 1)=f(an)=x/√( 1+nx^2)/√( 1+x^2/( 1+nx^2))

=x/√( 1+nx^2)/√(( 1+(n+ 1)x^2)/( 1+nx^2))

= x/√ ( 1+(n+ 1) x 2)。