Explicație pas cu pas:
[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]= \underbrace{\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + ... + \frac{1}{5}}_{80} = \frac{80}{5} = 16 = \bf {4}^{2}\\ [/tex]
q.e.d.
