Question

cele doua exercitii, dau coroana si 50 de puncte​

Answer 1

Explicație pas cu pas:

$a = \frac{1}{1\cdot 2} + \frac{1}{2 \cdot 3} + ... + \frac{1}{n \cdot (n + 1)} = \\ = \frac{1}{1} - \not\frac{1}{2} + \not\frac{1}{2} - \not\frac{1}{3} + ... + \not\frac{1}{n} - \frac{1}{n + 1} \\ = \frac{1}{1} - \frac{1}{n + 1} = \frac{n + 1 - 1}{n + 1} = \bf \frac{n}{n + 1}$

$n \geqslant 1 \implies 0 < \frac{n}{n + 1} \leqslant 1 \\ \implies 0 < a \leqslant 1$

$\Big\{a \Big\} = \frac{n}{n + 1} \implies \frac{n}{n + 1} = 0.999 \\ \frac{n}{n + 1} = \frac{999}{1000} \iff \frac{n}{n + 1} = \frac{999}{999 + 1} \\ \implies \bf n = 999$