## 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 ${2m+1 \choose m} \leq 2^{2m}$.

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