Show that any positive integral power of (√2 - 1) can be written in the form √N -
√(N-1) , where N is a positive integer.
Hint: Use mathematical induction and consider separately the odd and even powers of (√2 - 1).
We need to prove the following statement.
Statement : For any positive n, (√2 - 1)ⁿ = √N - √(N-1) for some positive integer N.
Note: N depends on n.
Before we prove the above statement, we will prove a theorem below.
Please see attachment for full question.
(sqrt(2) - 1)^n = sqrt(2a^2) - ...
This is a proof regarding integral powers.