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Question Number 107178 by 175mohamed last updated on 09/Aug/20

Commented by mr W last updated on 09/Aug/20

$$=\frac{\sqrt{3}-1}{2}+\frac{\sqrt{5}-\sqrt{3}}{2}+...+\frac{\sqrt{169}-\sqrt{167}}{2}$$$$=\frac{\sqrt{169}-1}{2}$$$$=6$$

Answered by Dwaipayan Shikari last updated on 09/Aug/20

$$\mathrm{First}\mathrm{term}$$$$\frac{1}{1+\sqrt{3}}=\frac{\sqrt{3}-1}{2}=\frac{\sqrt{3}}{2}-\frac{1}{2}$$$$2\mathrm{nd}\mathrm{term}=\frac{1}{\sqrt{3}+\sqrt{5}}=\frac{\sqrt{5}-\sqrt{3}}{2}$$$$...$$$$\frac{1}{\sqrt{167}+\sqrt{169}}=\frac{\sqrt{169}-\sqrt{167}}{2}$$$$\frac{\sqrt{3}}{2}-\frac{1}{2}+\frac{\sqrt{5}}{2}-\frac{\sqrt{3}}{2}+....\frac{\sqrt{167}}{2}-\frac{\sqrt{165}}{2}+\frac{\sqrt{169}}{2}-\frac{\sqrt{167}}{2}$$$$=6$$$$$$

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