Question

aflati nr. n natural
2+2^2+2^3+...+2^n=2^55-2

Answer 1

[tex]S=2^{1}+2^{2}+2^{3}+......+2^{n}\\2*S=2^{1}*2+2^{2}*2+2^{3}*2+.......+2^{n}*2\\ 2*S=2^{1+1}+2^{2+1}+2^{3+1}+....+2^{n+1}\\2*S=2^{2}+2^{3}+2^{4}+......+2^{n+1}\\ 2*S=(2^{1}+2^{2}+2^{3}+...+2^{n})+2^{n+1}-2^{1}\\Asadar:\\2*S=S+2^{n+1}-2^{1}\\De.unde:\\ S=2^{n+1}-2\\S-2=2^{n+1}\\n+1=55\\n=55-1\\n=54\\Sfarsit![/tex]