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:
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
Then I've evaluated it with the summation form
And confirmed that it's equal to 11 (I guess this is the base step).
The I did the same for
( 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.