Welcome 2016! Part 38

Algebra Level 2

1 1 + 2 + 1 2 + 3 + ⋯ + 1 7 + 8 + 1 8 + 9 = ? \large \dfrac{1}{1+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{7}+\sqrt{8}}+\dfrac{1}{\sqrt{8}+\sqrt{9}} = \, ?

∞ \infty 2 1 0 2 \sqrt{2}

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Anish Harsha by Anish Harsha , 20, India

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2 solutions

Rishabh Jain
Jan 12, 2016

1 1 + 2 = 2 − 1 \dfrac{1}{1+\sqrt{2}}=\sqrt{2}-1 1 2 + 3 = 3 − 2 \dfrac{1}{\sqrt{2}+\sqrt{3}}=\sqrt{3}-\sqrt{2} … \ldots ( 2 − 1 ) + ( 3 − 2 ) + ( 4 − 3 ) … ( 9 − 8 ) (\color{#D61F06}{\sqrt{2}}-1)+(\color{#3D99F6}{\sqrt{3}}\color{#D61F06}{-\sqrt{2}})+(\color{#20A900}{\sqrt{4}}\color{#3D99F6}{-\sqrt{3}})\ldots (\sqrt{9} \color{#EC7300}{-\sqrt{8})} = 9 − 1 = 2 \Large =\sqrt{9}-1= 2

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Nikhesh Kumar
Jul 17, 2016

Rationalize the denominators and simplify them.you will get √9-1=3-1=2.so 2 is the correct answer to this question!!

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