How can I show that: [tex]\sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2^{n}} [/tex] for every natural numbers
Well, this [tex] \sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2 n} [/tex] is an identity it is true for all [tex]n[/tex] but, if I understand correctly, you may ask for the values of [tex]n[/tex] that make [tex] \sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2^{n}} [/tex] true. In this case we have the equation [tex]2n=2^{2^{n}}[/tex], and the solutions are [tex]n \in \lbrace 1,2 \rbrace[/tex].
Version of AtomSeven is good. How can I show that [tex] \sum_{i=0}^{n}2^{n-i} {n+i \choose i}=2^{2 n} [/tex] is good for every natural numbers