[tex]S=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+...+\frac{1}{\sqrt{63}+\sqrt{64}}\\
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Vom amplifica fiecare fractie cu conjugatul numitorului.
Daca numitorul este (a + b), atunci vom amplifica cu (a - b), si vom folosi formula diferentei de patrate:
(a + b)(a - b) = a² - b²
[tex]S=\frac{\sqrt{2}-1}{(\sqrt{2}+1)(\sqrt{2}-1)}+\frac{\sqrt{3}-\sqrt{2}}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}+...+\frac{\sqrt{64}-\sqrt{63}}{(\sqrt{64}+\sqrt{63})(\sqrt{64}-\sqrt{63})}\\\\
S= \frac{\sqrt{2}-1}{2-1} + \frac{\sqrt{3}-\sqrt{2}}{3-2} +...+ \frac{\sqrt{64}-\sqrt{63}}{64-63}\\\\
S=\frac{(\sqrt{2}-1)+(\sqrt{3}-\sqrt{2})+(\sqrt{4}-\sqrt{3})+...+(\sqrt{64}-\sqrt{63})}{1}\\\\
S=-1+\sqrt{2}-\sqrt{2}+\sqrt{3}-\sqrt{3}+\sqrt{4}-\sqrt{4}+...-\sqrt{63}+\sqrt{64}\\\
S=\sqrt{64}-1=7
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