Explicație pas cu pas:
[tex]\frac{1}{n(n + 1)} = \frac{n + 1 - n}{n(n + 1)} = \frac{n + 1}{n(n + 1)} - \frac{n}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1} \\ \\ \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + ... + \frac{1}{n(n + 1)} = \frac{3 - 2}{2 \times 3} + \frac{4 - 3}{4 \times 3} + \frac{5 - 4}{5 \times 4} + ... + \frac{1}{n} - \frac{1}{n + 1} \\ \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + ... + \frac{1}{n} - \frac{1}{n + 1} = \frac{1}{2} - \frac{1}{n + 1} = \frac{n - 1}{2(n + 1)} = \frac{999}{2000} \\ 1000n - 1000 = 999n + 999 \\ n = 1999 [/tex]
La 5 e simplu acolo incerci formula radicalilor compusi pentru cel de al doilea mare radical
vei avea (V(5/2)-V(3/2))^2
si avem (V5-V3)/V2 inmultit cu prima chestie si deci (5-3)/V2=2V2/2=V2
Simplu nu?
