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!
Prove that:
(i) The inequality is true for $\displaystyle k=4$ .
(ii) If it is true for $\displaystyle k\geq 4$ integer, then it is true for $\displaystyle k+1$ .
Fernando Revilla
$\displaystyle kk-3k\ge\ 4\Rightarrow\ k(k-3)\ge\ 4$
$\displaystyle k$ cannot be 3 or less.
$\displaystyle k=4\Rightarrow\ 4(1)=4$
Then we have that the inequality ought to be true for $\displaystyle k\ge\ 4$
P(k)
$\displaystyle k(k-3)\ge\ 4$
P(k+1)
$\displaystyle (k+1)(k-2)\ge\ 4$
Try to show that "if" P(k) is true, "then" P(k+1) will also be true
(establish the inductive "cause and effect")
Proof
$\displaystyle (k+1)(k-2)=(k+1)(k-2-1)+(k+1)=(k+1)(k-3)+(k+1)$
$\displaystyle =k(k-3)+(k-3)+(k+1)=k(k-3)+2k-2$
If P(k) is true, then the above is $\displaystyle \ge\ 4+2k-2$
with $\displaystyle k\ge\ 4$
and so P(k+1) is true also.