Prove Binomial Theory by Mathematic Induction

Cn,r<(n+1)^r ;0<=r<=n

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- Dec 16th 2009, 03:23 AMcalfeverProve Binomial Theory
Prove Binomial Theory by Mathematic Induction

Cn,r<(n+1)^r ;0<=r<=n - Dec 16th 2009, 03:24 AMmr fantastic
Use Google.

proof binomial theorem induction - Google Search= - Dec 16th 2009, 11:02 AMawkward
I think Mr. F has misinterpreted the problem statement due to its misleading title. I think the problem is:

Prove that

for all ....(*)

by mathematical induction.

Note that I have changed the less-than in the original statement to less-than-or-equal. Otherwise (*) would be false when r=0.

I'm going to assume you know the important identity .

First check the case :

If then the only possible value of r is .

and , so (*) is true.

Now suppose that for some integer k we have

for all .

I will let you check the case .

If then , so by assumption we also have

.

Adding the two inequalities,

so by the identity stated above,

so (*) holds for

By mathematical induction, (*) is true for all . - Dec 16th 2009, 01:59 PMcalfever
thanks a lot!! AWKWARD