#### What should be proved in the binomial theorem?

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User

( 4 months ago )

I'm following Cambrige mathematics syllabus, from the list of contents of what should be learned:

Induction as a method of proof, including a proof of the binomial theorem with non-negative integral coefficients.

I know what it is, but I'm not sure of what should be proved here. The first thing that comes to mind is the idea of proving it it for n+1n+1, but I thought about writing:

• (a+b)n(a+b)n

• (a+b)n+1(a+b)n+1

But I am missing what premise I should prove. I guess that the proof involves the nature of the coefficients of the expansion of (a+b)n(a+b)n but from here, I have no idea on how to proceed. Can you help me?

Edit: I guess I've made some progress. First I evaluated

(x+y)0=1(x+y)0=1

Then I've evaluated it with the summation form

j=00(nj)xnjyj∑j=00(nj)xn−jyj

And confirmed that it's equal to 11 (I guess this is the base step).

The I did the same for

User

( 4 months ago )

If you know the expansion of (a+b)n(a+b)n, it follows that the expansion of (a+b)n+1=(a+b)(a+b)n=a(a+b)n+b(a+b)n(a+b)n+1=(a+b)(a+b)n=a(a+b)n+b(a+b)n can be found by distributing term by term and collecting coefficients. This is the type of reasoning you should use when doing inductive proofs.

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