# principle of mathematical induction

• May 2nd 2012, 08:09 AM
qwerty31
principle of mathematical induction
How do I prove by induction that n^2 is greater than or equal to 3n + 4 for n > 3?
• May 2nd 2012, 08:43 AM
a tutor
Re: principle of mathematical induction
First prove it for n=4.

Next show that if it's true n=k then it's true for n=k+1.

But I suppose you know that already...so why not say how far you got and where you're stuck.
• May 2nd 2012, 09:01 AM
qwerty31
Re: principle of mathematical induction
Quote:

Originally Posted by a tutor
First prove it for n=4.

Next show that if it's true n=k then it's true for n=k+1.

But I suppose you know that already...so why not say how far you got and where you're stuck.

I've gotten to 'assume true for n = k' and I dont know how to go about proving that its true for k + 1
• May 2nd 2012, 09:16 AM
a tutor
Re: principle of mathematical induction
Well, if it's true for n=k then \$\displaystyle k^2\ge 3k+4\$

What does that tell you about \$\displaystyle (k+1)^2\$?
• May 2nd 2012, 09:18 AM
qwerty31
Re: principle of mathematical induction
Quote:

Originally Posted by a tutor
Well, if it's true for n=k then \$\displaystyle k^2\ge 3k+4\$

What does that tell you about \$\displaystyle (k+1)^2\$?

(k+1)^2 is greater than or equal to 3(k+1) + 4?
• May 2nd 2012, 12:53 PM
a tutor
Re: principle of mathematical induction
OK, \$\displaystyle (k+1)^2=k^2+2k+1\$.

Now, since \$\displaystyle k^2\ge3k+4\$

we have

\$\displaystyle (k+1)^2=\$

\$\displaystyle k^2+2k+1\ge 5k+5=3(k+1)+2k+2= 3(k+1)+4+2k-2\ge 3(k+1)+4\$
• May 24th 2012, 04:37 AM
kalwin
Re: principle of mathematical induction
The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
1. The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1.
2. The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.

The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.
• May 24th 2012, 05:03 AM
skeeter
Re: principle of mathematical induction
Quote:

Originally Posted by kalwin
The simplest and most common form of mathematical induction proves that a statement involving a natural number n holds for all values of n. The proof consists of two steps:
1. The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1.
2. The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.

The assumption in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.

next time, just provide the link ...

2.1.1 Description