Question

calculati suma s=1/2-1/2 la 2 +1/2 la 3-1/2 la 4+....+1/2 la 19-1/2 la 20

Answer 1

Răspuns:

1/3-1/3*2^21= (1/3) (1-1/2^21)

sau, altfel scris,

(2^21-1)/3*2^21

Explicație pas cu pas:

calculam

S=1-s=1-1/2+1/2²-1/2³+......-1/2^19+1/2^20= (1-(-1/2)^21)/(1-(-1/2)**=

(1+1/2^21)/(3/2)= (2/3)*(2^21+1)/2^21= 2(2^21+1)/3*2^21

s=1-S=1-(1-s)

s= 1-  2(2^21+1)/3*2^21=

=(3*2^21-2*2^21-1)/3*2^21=

=(2^21-1)/3*2^21= eventual  1/3-1/3*2^21= (1/3) (1-1/2^21)

**** S20 pt progresie geometrica, avand b1=1 si q=-1/2