Question

Aratati ca 2 ^{0} + 2^{1} + 2^{2} + 2^{3} +...+ 2^{50} = 2^{51}-1.

Answer 1

[tex]2^{0}+2^{1}+2^{2}+2^{3}+.....+2^{50}=2^{50+1}-1=2^{51}-1\\Am\ aplicat\ formula:\\ 2^{0}+2^{1}+2^{2}+2^{3}+......+2^{n}=2^{n+1}-1\\Care\ se\ demonstreaza:\\ S=2^{0}+2^{1}+2^{2}+......+2^{n}\\Inmultind\ cu\ 2,\ obtinem:\\ 2*S=2*(2^{0}+2^{1}+2^{2}+....+2^{n})\\2*S=2*2^{0}+2*2^{1}+2*2^{2}+.......+2*2^{n}\\ 2*S=2^{1}*2^{0}+2^{1}*2^{1}+2^{1}*2^{2}+.........+2^{1}*2^{n}\\ 2*S=2^{1+0}+2^{1+1}+2^{2+1}+......+2^{n+1}\\2*S=2^{1}+2^{2}+2^{3}+2^{n+1}\\ 2*S=(2^{0}+2^{1}+2^{2}+....+2^{n})+2^{n+1}-2^{0}\\2*S=S+2^{n+1}-2^{0}\\
2*S-S=2^{n+1}-2^{0}\\S=2^{n+1}-1[/tex]