principle of mathematical induction

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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 $k^2\ge 3k+4$

What does that tell you about $(k+1)^2$ ?
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 $k^2\ge 3k+4$

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

(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, $(k+1)^2=k^2+2k+1$ .

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

we have

$(k+1)^2=$

$k^2+2k+1\ge 5k+5=3(k+1)+2k+2= 3(k+1)+4+2k-2\ge 3(k+1)+4$
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:

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.
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.
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