Question

Aratati ca : 1/n-1/n+1=1/n(n+1) si folosind aceasta formula calculati:
a) S=1/1*2+1/2*3+1/3*4+...+1/89*90
b) S= 1/6*7+1/7*8+1/8*9+...+1/74*75
c) S= 1/4*5+1/5*6+1/6*7+...+1/71*72
d) S= 1/7*8+1/8*9+1/9*10+...+1/90*91

Answer 1

1/n - 1/(n+1) = (n+1-n)/n(n+1) = 1/n(n+1)
a)  Sa = (1/1 - 1/2) + (12-1/3) + (1.3-1/4) + ........+ (1/89 - 1/90) =
= 1/1 - 1/90 = 89/90
b)  Sb = (1/6 - 1/7) + (1/7 - 1/8) + (1/8 - 1/9) +........+(1/74 - 1/75) =
= 1/6 - 1/75 = (25 - 2)/150 = 23/150
c)  Sc = (1/4-1/5) + (1/5-1/6) + (1/6 - 1/7) +........+ (1/74 - 1/75) =
= 1/4 - 1/75 = (75 - 4) /300 = 71/300
d)  Sd = (1/7 - 1/8) + (1/8 - 1/9) + .....+ (1/90 - 1/91) =
= 1/7 - 1/91 = (13 - 1)/91 = 12/91