Răspuns:
a,1/2ⁿ⁻¹-1/2ⁿ=
1/2ⁿ⁻¹-1/2*2ⁿ⁻¹=(2-1)/2*2ⁿ⁻¹=1/2ⁿ
b)1/2-(1/2²+1/2³+1/2⁴+...+1/2¹⁰)=
In paranteza ai o progresie geometrica cu 9 termeni cu ratia1/2 si primul termen 1/2²
deci calculezi suma dupa formula
Sn=a1*(qⁿ-1)/(1-q) unde a1=1/2² q=1/2. n=9
S=1/2²[(1/2)⁹-1]/(1-1/2)=
1/4[(1/2)⁹-1]/(1/2)=
[(1/2)⁹-1]/(4/2)=[(1/2)⁹-1]/2
1/2-[(1/2)⁹-1]/2=
[1-(1/2)⁹-1]/2=
(1/2)⁹/2=1/2¹⁰
Explicație pas cu pas:
