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

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- May 2nd 2012, 08:09 AMqwerty31principle 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 AMa tutorRe: 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 AMqwerty31Re: principle of mathematical induction
- May 2nd 2012, 09:16 AMa tutorRe: 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 AMqwerty31Re: principle of mathematical induction
- May 2nd 2012, 12:53 PMa tutorRe: 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 AMkalwinRe: 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:

- 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 thatthe statement holds for some**if***n*,the statement also holds when**then***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.

- The
- May 24th 2012, 05:03 AMskeeterRe: principle of mathematical induction
next time, just provide the link ...

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