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## Using Binomial Theorem

Prove the following using binomial theorem and/or mathematical induction

Let \(a\), \(b\) and \(n\) be natural numbers, prove that \(\frac{(a+\sqrt{b})^n + (a-\sqrt{b})^n}{2}\) is also a natural number.

Let m be a non-negative integer. Prove that \({2m+1 \choose m} \leq 2^{2m}\).