Question

Use the binomial theorem to prove that

C(n,0)+C(n,1)2+C(n,2)2^2 +⋯+C(n,n)2^n =3^n

Answer 1

i kn the rows of the pascal triangle is 2^n

Answer 2

but i thought C(n,n) was just equal to one

Answer 3

it is

Answer 4

so how is 2^n=3^n

Answer 5

only the last term is \(2^n\)

Answer 6

i have no idea how to do this, but it says "use the binomial theorem" so that has to be a clue

Answer 7

may be has something to do with
\[(1+2)^n=3^n\]?

Answer 8

that looks promising actually

Answer 9

thats what am thinking ... but am trying to do it use algebra to get it to it

Answer 10

expand
\[(1+2)^n\] using the binomial theorem and i think you get it

Answer 11

first term is
\[\binom{n}{0}1^n2^0=\binom{n}{0}\]
second term is
\[\binom{n}{1}1^{n-1}2^1=\binom{n}{1}2\]

Answer 12

etc
it works out to exactly what you want

Answer 13

that seems to work