Find 11+2+12+3+13+4++180+81.

GATE 2013 CS (CS1) – Question 6

Find $\frac1{1+\sqrt2}+\frac1{\sqrt2+\sqrt3}+\frac1{\sqrt3+\sqrt4}+\cdots+\frac1{\sqrt{80}+\sqrt{81}}$.

Show answer and explanation

Correct answer: (B) 8

Rationalizing each denominator gives $1/(\sqrt k+\sqrt{k+1})=\sqrt{k+1}-\sqrt k$. The sum telescopes from k=1 to 80, leaving √81−√1=9−1=8, B.

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