Explicație pas cu pas:
35) știm că:
[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]
înmulțim, pe verticală:
[tex]\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} ... \cdot \frac{49}{50} < \frac{2}{3} \cdot \frac{4}{5} \cdot \frac{6}{7} ... \cdot \frac{50}{51} < \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} ... \cdot \frac{51}{52} \\ [/tex]
36)
[tex]a = \frac{1}{5} + \frac{7}{10} + \frac{8}{15} + ... + \frac{85}{400} - ( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{80}) = \\ [/tex]
[tex]= \frac{1}{5} + \left( \frac{7}{2 \cdot 5} - \frac{1}{2} \right) + \left( \frac{8}{3 \cdot 5} - \frac{1}{3}\right) + ... + \left( \frac{85}{80 \cdot 5} - \frac{1}{80} \right) \\ [/tex]
[tex]= \frac{1}{5} + \frac{7 - 5}{2 \cdot 5} + \frac{8 - 5}{3 \cdot 5} + ... + \frac{85 - 5}{80 \cdot 5} \\ [/tex]
[tex]= \frac{1}{5} + \frac{2}{2 \cdot 5} + \frac{3}{3 \cdot 5} + ... + \frac{80}{80 \cdot 5} \\ [/tex]
[tex]= \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + ... + \frac{1}{5} = \frac{80}{5} = 16 = \red{\bf {4}^{2}} \\ [/tex]
q.e.d.
