Question

a ) Verificati ca : $\frac{1}{1 x 2}$ = $\frac{1}{1}$-$\frac{1}{2}$ si$\frac{1}{2x3}$ = $\frac{1}{2}$-$\frac{1}{3}$.
b ) Aratati cu S = $\frac{1}{1 ori 2}$ =$\frac{1}{2 ori 3}$+ ... +$\frac{1}{9ori10}$=$\frac{9}{10}$.Completati suna S cu termeni asemanatori astfel incat sa fie egala cu $\frac{99}{100}$ ! Va rogggg !Unde este x este ori ! CORONITA

Answer 1

1/ 1*2=1/1-1/2=2/2-1/2  adevarat

1/ 2*3= 1/2-1/3=3/6-2/6= 1/6  adevarat

S= 1/ 1*2 + 1/ 2*3+......+1/ 9*10= 9/10

S= 1/2 +1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 +1/90 = 9/10.

2=2
6=2*3
12=2²*3
20=2²*5
30=2*3*5
42=2*3*7
56=2³*7
72=2³*3²
90=2*3²*5
cmmmc=2³*3²*5*7=8*9*5*7=72*35=2520

S=1260/2 520 + 420/2 520 + 210/2520 + 126/2520 + 84/2520 + 60/ 2520 + 45/2520 + 35/2520 + 28/2520 =2268/2520   /:2
S=1134/1260   /:2
S=567/630   /:3
S=189/210    /:3
S= 63/70   / :7
S= 9/10

S= 1/1*2+1/2*3+1/3*4+1/3*5..........+1/9*10.....+1/ 99*100 = 99/100