Inequalities and mathematical induction: RMO Homi Bhabha sample questions:

Prove by mathematical induction, the following:

1) $2^{n}(n!)^{2} \leq (2n)!$ for all $n \geq 1$ .

2) Establish the Bernoulli inequality: If $(1+a)>0$ , then $(1+a)^{n} \leq 1+na$ for all natural numbers greater than or equal to 1.

3) For all $n \geq 1$ with $n \in N$ prove the following by mathematical induction:

a) $\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \ldots + \frac{1}{n^{2}} \leq 2-\frac{1}{n}$

Solutions will be put up tomorrow!

Nalin Pithwa.

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