Question

0,8 (3)<1/1×2 + 1/2×3 + 1/3×4 +...+1/n (n+1) < 0,91 (6) cat este n

Answer 1

1/1×2 =1/1-1/2=(2-1)/1×2=1/1×2
1/2×3 =1/2-1/3=(3-2)/2×3=1/2×3
1/3×4 =1/3-1/4=(4-3)/3×4=1/3×4   
...
1/n(n+1)=1/n-1/(n+1)=(n+1-n)/n(n+1)

1/1×2 + 1/2×3 + 1/3×4 +...+1/n (n+1) =1/1-1/2+1/2-1/3+1/3-1/4+...+1/n-1/(n+1)=
=1/1 -1/(n+1)=(n+1-1)/(n+1)=n/(n+1)


0,8 (3)<n/(n+1) < 0,91 (6)
0,8(3)=(83-8)/90=75/90=5/6
0,91 (6)=(916-91)/900=825/900=11/12
5/6<n/(n+1) < 11/12
n={6,7,8,9,10}