Question

35 Arătaţi că:
1/2 x 3/4 x 5/6 x 7/8 x … x 49/50 < 2/3 x 4/5 x 6/7 x … x 50/51 < 3/4 x 5/6 x 7/8 x … x 51/52
Va rog repede!

Answer 1

Explicație pas cu pas:

$\boxed{ \bf\frac{n}{n + 1} < \frac{n + 1}{n + 2}}$

atunci:

$\frac{1}{2} < \frac{2}{3} < \frac{3}{4} \\ \frac{3}{4} < \frac{4}{5} < \frac{5}{6} \\ \frac{5}{6} < \frac{6}{7} < \frac{7}{8} \\ .... \frac{49}{50} < \frac{50}{51} < \frac{51}{52}$

înmulțim membru cu membru (pe verticală):

$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot ... \cdot \frac{49}{50} < \frac{2}{3} \cdot \frac{4}{5} \cdot \frac{6}{7} \cdot ... \cdot \frac{50}{51} < \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot ... \cdot \frac{51}{52}$

q.e.d.