Binomial Theorem

it says to use binomial theorem to prove:

(n / 0) - (n / 1) + (n / 2) - ... + (-1)^n (n / n) = 0

for this one , i know that from the first term a = 1 but i don't know how to do the rest

(n / 0) + 2(n / 1) + 4 (n / 2) + 4 (n /2) + 8 (n 3) +... + 2^n( n /n ) = 3^n

for this one, a = 1 and if i plug in 1 for second term, i get b= 2 but i dont know how to prove it

jijie14 said:

it says to use binomial theorem to prove:
(n / 0) - (n / 1) + (n / 2) - ... + (-1)^n (n / n) = 0

Click to expand...

@jijie14, You have real problems with notation.

\(\displaystyle \left( {x + y} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}x^k y^{n - k} } \) is the binomial theorem.

Now let \(\displaystyle x=-1~\&~y=1\) so you get \(\displaystyle 0=\left( {-1 + 1} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}(-1)^k (1)^{n - k} } \)