Induction Inequality

**jstarks44444** · February 17th 2011, 10:37 AM

Any help with the following proof would be appreciated,

Determine for which natural numbers k^2 - 3k >= 4 and prove your answer.

How would we use induction here? Thanks in advance for all the help!

**FernandoRevilla** · February 17th 2011, 10:46 AM

Prove that:

(i) The inequality is true for $k=4$ .

(ii) If it is true for $k\geq 4$ integer, then it is true for $k+1$ .

Fernando Revilla

**Archie Meade** · February 17th 2011, 01:47 PM

Originally Posted by jstarks44444

Any help with the following proof would be appreciated,

Determine for which natural numbers k^2 - 3k >= 4 and prove your answer.

How would we use induction here? Thanks in advance for all the help!

$kk-3k\ge\ 4\Rightarrow\ k(k-3)\ge\ 4$

$k$ cannot be 3 or less.

$k=4\Rightarrow\ 4(1)=4$

Then we have that the inequality ought to be true for $k\ge\ 4$

P(k)

$k(k-3)\ge\ 4$

P(k+1)

$(k+1)(k-2)\ge\ 4$

Try to show that "if" P(k) is true, "then" P(k+1) will also be true
(establish the inductive "cause and effect")

Proof

$(k+1)(k-2)=(k+1)(k-2-1)+(k+1)=(k+1)(k-3)+(k+1)$

$=k(k-3)+(k-3)+(k+1)=k(k-3)+2k-2$

If P(k) is true, then the above is $\ge\ 4+2k-2$

with $k\ge\ 4$

and so P(k+1) is true also.

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