# Induction Mathematical Proof and Odd and Even Powers

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Show that any positive integral power of (âˆš2 - 1) can be written in the form âˆšN -

âˆš(N-1) , where N is a positive integer.

Hint: Use mathematical induction and consider separately the odd and even powers of (âˆš2 - 1).

We need to prove the following statement.

Statement : For any positive n, (âˆš2 - 1)ⁿ = âˆšN - âˆš(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.

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#### Solution Preview

Because,

(sqrt(2) - 1)^n = sqrt(2a^2) - ...

#### Solution Summary

This is a proof regarding integral powers. The induction mathematical proof and odd and even powers.

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