Use binomial theorem to prove

[ND3G]

Use binomial theorem to prove

C(n,0) - 3(C(n,1)) + 9(C(n,2) - 27(C(n,3) + ... + (-3)^n(C(n,n) = (-2)^n

From looking at the data given b = (-3) so a = 1 so (-2)^n = (1-3)^n

With this I know the equation in sigma notation and could probably prove the theorem through mathematical induction but I'm not certain that is what they are looking for in this case...

Update:

I proved n=1 and assumed n = k, so do I need to prove n= k+1 through mathematical induction?

 

[Vagrant]

Doesn't writing out the expression for $$(1+x)^n$$ where x=-3 constitute proof?

edit: it's $$(1+x)^n$$

 

[Architeuthis Dux]

Yes, in order to prove the binomial theorem, you will need to use mathematical induction to prove the statement for all values of n. This means that you will need to prove the statement for n = 1 and then assume that it is true for some arbitrary value k, and prove that it is also true for n = k+1. This will allow you to show that the statement holds true for all values of n.