1. ## binomial theorem questions.

Can anyone please show me the the proof of the ff?

The summation of (nCr)2^r <as r goes from 0 to n> = [3^n + (-1)^n]/2

The summation of r(nCr) <as r goes from 1 to n>= n(2^n-1)
(combinatorial proof)

Your help would be very much appreciated. Thanks.

2. Don't you mean

[LaTeX ERROR: Compile failed]

or

[LaTeX ERROR: Compile failed]

These two results follow directly from the application of the binomial theorem in reverse.

[LaTeX ERROR: Compile failed]

with x = 1 and y = 2 where in the second one we are subtracting 1 since we are not summing over the first term which is [LaTeX ERROR: Compile failed]

EDIT: Here is the second one:

[LaTeX ERROR: Compile failed]

[LaTeX ERROR: Compile failed]

Now the sum becomes:

[LaTeX ERROR: Compile failed]