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.

Enter your comment here...

Fill in your details below or click an icon to log in:

Email (required) (Address never made public)

Name (required)

Website

You are commenting using your WordPress.com account. ( Log Out / Change )

You are commenting using your Google account. ( Log Out / Change )

You are commenting using your Twitter account. ( Log Out / Change )

You are commenting using your Facebook account. ( Log Out / Change )

Connecting to %s

%d bloggers like this: