Binomial Theorem - Proof
Show that, for n > 0,
I know i am supposed to use the binomial theorem which is just,
I am really just not sure where to start though :(
Thank you for any help/tips :)
(if this needs to be moved to Discrete Mathematics, Set Theory and Logic, would a mod please move it. thank you)
Here is a different hint.
That is of course a sum.
Or, you could let
or something similar and plug them into the expansion, noticing a pattern.
Ok great i think i got it.
for a = 1 and b = -1 and using any number where n > 0 you can see that when plugged into the binomial theorem it is a repeating pattern of (in this case) 2 + -2 + 2 + -2 .... which therefore will = 0 proving to be true.
Does this seem correct or am i missing something? I am very new to proofs (this is actually my first one haha :) )
I have been looking over these posts again for the past day or so and I still can't seem to wrap my mind around this.
I get using A = -1 and B = 1 and using the binomial theorem will give you the expansion
I guess i just dont get how this proves that for n > 0
I feel like i am making this much more difficult than it should be :(
in each term of the expansion.
TheCoffeeMachine related the binomial expansion to your closed-form sum.
They are the same.
Hence there is no need to calculate each term of the expansion, since the sum is
as Plato also showed.
Thank you for the great explanation :D I am going to give my brain a rest then rework the proof and make sure i understand everything and if I dont I will post back again :D
Thanks again for all the replies everyone. I really appreciate the help
You said you are supposed to use the binomial theorem. So for we have,
let a = -1 and b = 1. This equality then becomes,
So the binomial theorem actually proves it for you. You just need to plug in those values. This is just a special case of the binomial theorem.
You could also use the binomial theorem to show that
in each case you are using the special cases of a theorem that has already been proven for you. I hope that helps you out a bit.