The kids all guessed it is two by successive approxmation:

√2 = 1.414

√(2 + √2) = √ 3.414 = 1.874

√(2 + √(2 + √2)) = √3.874 = 1.9615

Algebraically we can write set *x*= √(2 + √(2 + √(2 + ...))). Then we observe that the inner expression is also *x*, so we have *x*=√(2+x).

Solving for *x* we get

*x*^{2} = 2 + *x*,

or

*x*^{2} - *x* -2 = 0.

You could use the quadratic equation or you could guess the factorization. Somehow, by hook or by crook you get

(*x*-2)(*x*+1)=0.

And so *x*=2, as the kids guessed.
The kids observed that it's the same trick for solving a geometric series:

1+1/2+1/4+1/8+ ... = *x*,

so

1 + *x*/2 = *x*,

which has solution *x*=2.

## No comments:

## Post a Comment