Question

Daca m/n este o valoare aproximativa a lui sqrt(3), aratati ca (m+3n)/(m+n) este o aproximatie "mai buna" a lui sqrt(3).
P.S: sqrt=square root=radacina patrata

Answer 1

$\frac{m}{n} = \sqrt{3}\Bigg|\cdot n\\ \\ m = n\sqrt{3}\\ \\ \frac{m+3n}{m+n} \\ \\= \frac{n\sqrt{3} + 3n}{n\sqrt{3} + n} \\ \\ = \frac{n(\sqrt{3} + 3)}{n(\sqrt{3}+1)} \\ \\ = \frac{\sqrt{3}+3}{\sqrt{3}+1} \\ \\ = \frac{\sqrt{3} + \sqrt{3}\cdot \sqrt{3}}{\sqrt{3}+1}\\ \\ = \frac{\sqrt{3}(1 + \sqrt{3})}{\sqrt{3}+1} \\ \\ = \frac{\sqrt{3}(\sqrt{3}+1)}{\sqrt{3}+1} \\ \\= \boxed{\sqrt{3}}$