Thread: Math Induction...PLEASE HELP!!!!!

Use math induction to show that
1) n!>= n^2 for n>=6

2) n!>=n^3 for n>=6

Originally Posted by xomichelleybelly

Use math induction to show that
1) n!>= n^2 for n>=6

I will assume that you have done the base case and that we assume this is true for some

So we need to show that:

,

well:



and by supposition , so:



So if we can prove that we are done, but is equvalent to . So to compete the proof of the induction step you need to show that:



has no real root greater than or equal to (this is sufficient as for greater than its largest real root the cubic is positive).

This you do by making the substitution , then using Descartes rule of signs to show that this has no positive roots.

(In fact the only real root of is near )

RonL

Originally Posted by xomichelleybelly

Use math induction to show that

2) n!>=n^3 for n>=6

Exactly the same method as for part 1, but the equation that you need to
show has no root greater than 6 is:



and exactly the same method will work with this.

RonL

Originally Posted by xomichelleybelly

Use math induction to show that
1) n!>= n^2 for n>=6

2) n!>=n^3 for n>=6

Halve your workload, chum: Start with 2).

By proving 2) you've automatically proved 1).

(And I reckon a lawyer could argue that induction has been used to prove both )