Question

Rezultatul calculului 1+2+2²+2³+.....2la puterea 99 impartit la 7 este:

Answer 1

$S=2^{0}+2^{1}+ 2^{2} +2^{3}+...+2^{99}|:2 \Rightarrow \\ \Rightarrow \frac{S}{2} = 2^{-1}+2^{0} +2^{1} + 2^{2} + 2^{3} +...+2^{98}+(2^{99} - 2^{99}) \Rightarrow \\ \Rightarrow \frac{S}{2} = 2^{-1} +S - 2^{99} \Rightarrow \frac{S}{2}-S = 2^{-1}-2^{99} \Rightarrow - \frac{S}{2} = 2^{-1} -2^{99} \Rightarrow \\ \Rightarrow -S = 2^0 - 2^{100} \Rightarrow S = 2^{100} - 1 \\ \\ \Rightarrow \frac{S}{7} = \frac{2^{100} - 1}{7}$

Deci, rezultatul este: 181 092 942 889 747 057 356 671 886 482

Answer 2

S=1+2+2²+2³+...+2⁹⁹ /·2
2S=2+2²+2³+...+2¹⁰⁰
S=2S-S
S=2¹⁰⁰-1
S/7=2¹⁰⁰-1 /7