# Thread: principle of mathematical induction

1. ## principle of mathematical induction

How do I prove by induction that n^2 is greater than or equal to 3n + 4 for n > 3?

2. ## 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.

3. ## Re: principle of mathematical induction

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

4. ## 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 $(k+1)^2$?

5. ## Re: principle of mathematical induction

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?

6. ## Re: principle of mathematical induction

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

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

we have

$(k+1)^2=$

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

7. ## 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.

8. ## Re: principle of mathematical induction

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