Explicație pas cu pas:
35. avem inegalitățile:
[tex]\frac{1}{2} < \frac{2}{3} < \frac{3}{4} \\ [/tex]
[tex]\frac{3}{4} < \frac{4}{5} < \frac{5}{6} \\ [/tex]
[tex]\frac{5}{6} < \frac{6}{7} < \frac{7}{8} \\ [/tex]
...
[tex]\frac{49}{50} < \frac{50}{51} < \frac{51}{52} \\ [/tex]
acum înmulțim relațiile între ele (pe verticală) și obținem:
[tex]\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} \\ [/tex]
36.
[tex]a = \frac{1}{5} + \frac{7}{10} + \frac{8}{15} + ... + \frac{85}{400} - \left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{80} \right) = \\ [/tex]
[tex] = \frac{1}{5} + \left(\frac{1}{2} + \frac{1}{5} \right) + \left(\frac{1}{3} + \frac{1}{5} \right) + \left(\frac{1}{4} + \frac{1}{5} \right) + ... + \left(\frac{1}{80} + \frac{1}{5} \right) - \left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{80} \right) \\ [/tex]
[tex]= 80 \cdot \frac{1}{5} + \left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{80} \right) - \left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{80} \right) \\ [/tex]
[tex]= \frac{80}{5} = 16 = {4}^{2} \\ [/tex]
q.e.d.
