Question

36 Arătaţi că numărul a Rezolvare: 8 5 10 15 85 400 234 80 este pătrat perfect​

Answer 1

Explicație pas cu pas:

$a = \frac{1}{5} + \frac{7}{10} + \frac{8}{15} + ... + \frac{85}{400} - ( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{80}) =$

$= \frac{1}{5} + \left( \frac{7}{2 \cdot 5} - \frac{1}{2} \right) + \left( \frac{8}{3 \cdot 5} - \frac{1}{3}\right) + ... + \left( \frac{85}{80 \cdot 5} - \frac{1}{80} \right)$

$= \frac{1}{5} + \frac{7 - 5}{2 \cdot 5} + \frac{8 - 5}{3 \cdot 5} + ... + \frac{85 - 5}{80 \cdot 5}$

$= \frac{1}{5} + \frac{2}{2 \cdot 5} + \frac{3}{3 \cdot 5} + ... + \frac{80}{80 \cdot 5}$

$= \underbrace{\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + ... + \frac{1}{5}}_{80} = \frac{80}{5} = 16 = \bf {4}^{2}$

q.e.d.