Question

calculati s=1-2+2^2-2^3+...+2^100.Rapid

Answer 1

progresie geomerica cu ratia -2
 a0* (q^(n+1)-1)/(q-1)= 1* (-2)^101-1)/(-2-1)=-2^101-1/(-3) =
 -(2^101+1)/(-3)=(1+2^101)/3= (2^101+1)/3

altfel
1-2=-1
4-8=-4=-2²
16-32=-16=-2^4
............
2^98-2^99=-2^98

2^100=+2^100


-(1+2²+2^4+...+2^98) +2^100=

-(1+2²+(2²)²+...+(2^49)²)+2^100=

-((2²)^50-1)/(2²-1)+2^100=

-(2^100-1)/3+2^100= (3*2^100-2^100+1)/3=(2*2^100+1)/3=(2^101+1)/3