Question

Determine a soma:
1+C(4n,1)+C(4n,2)+...+(4n,4n-2)+C(4n,4n-1)+1 onde n ∈ N

Answer 1

Recordando el binomio de Isaac Newton

$\displaystyle (a+b)^n=\sum\limits_{i=0}^{n} \binom{n}{i}a^{n-i}b^i\\ \\ \text{Ent\~ao: }\\ \\ (1+1)^{4n}=\sum\limits_{i=0}^{4n} \binom{4n}{i}(1)^{4n-i}1^i\\ \\ 2^{4n}=\sum\limits_{i=0}^{4n} \binom{4n}{i}\\ \\ \\ \boxed{1+\binom{4n}{1}+\binom{4n}{2}+\cdots +\binom{4n}{4n-1}+1=16^n}$