Question

Aratati ca..........​

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

Explicație pas cu pas:

a)

$\frac{1}{8} < \frac{1}{3} \leqslant \frac{1}{3} \\ \frac{1}{8} < \frac{1}{4} < \frac{1}{3} \\ \frac{1}{8} < \frac{1}{5} < \frac{1}{3} \\ \frac{1}{8} < \frac{1}{6} < \frac{1}{3} \\ \frac{1}{8} < \frac{1}{7} < \frac{1}{3} \\ \frac{1}{8} \leqslant \frac{1}{8} < \frac{1}{3}$

adunăm:

$6 \cdot \frac{1}{8} < \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} < 6 \cdot \frac{1}{3} \\ \implies \frac{3}{4} < \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} < 2$

b)

$\frac{1}{27} < \frac{1}{10} \leqslant \frac{1}{10} \\ \frac{1}{27} < \frac{1}{11} < \frac{1}{10} \\ \frac{1}{27} < \frac{1}{12} < \frac{1}{10} \\ .... \\ \frac{1}{27} \leqslant \frac{1}{27} < \frac{1}{10}$

între 10 și 27 sunt: 27-10+1 = 18 termeni

adunăm:

$18 \cdot \frac{1}{27} < \frac{1}{10} + \frac{1}{11} + \frac{1}{12} + ... + \frac{1}{27} < 18 \cdot \frac{1}{10} \\ \implies \frac{2}{3} < \frac{1}{10} + \frac{1}{11} + \frac{1}{12} + ... + \frac{1}{27} < \frac{9}{5}$

c)

$a < b \ \ \ | ( + ab) \iff a + ab < b + ab \\ a(b + 1) < b(a + 1) \implies \frac{a}{a + 1} < \frac{b}{b + 1}$

d)

$a < b \ \ \ | ( \cdot 2013 \cdot c) \iff \\ a\cdot 2013 \cdot c < b\cdot 2013 \cdot c \ \ \ | ( + ab) \iff \\ a\cdot 2013 \cdot c + ab < b\cdot 2013 \cdot c + ab \\ a(b + 2013 \cdot c) < b(a + 2013 \cdot c) \\ \implies \frac{a}{b} < \frac{a + 2013 \cdot c}{b + 2013 \cdot c}$

q.e.d.