Find the value of 1 1+ 2 + 1 2+ 3 + \cdots + 1 1599+ 1600.

Free Practice Question

PS · Problem Solving

605-655

Roots

Find the value of $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \cdots + \frac{1}{\sqrt{1599}+\sqrt{1600}}$ .

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Comments

NoelleJul 31

Rationalizing each term makes this a telescoping sum, super clean once you see it.

AbhilashaAug 13

I always forget rationalizing is the move here. Thanks for the nudge.

PurvaJun 19

Wait, do we rationalize every single term or is there a shortcut with the pattern? I started multiplying numerator and denominator by the conjugate but got lost around 3 terms in.

GabrieleJun 30

Conjugate is the way. Each fraction becomes the difference of two consecutive square roots, then everything cancels.

RishikJul 22

The 1599 and 1600 are a dead giveaway that it telescopes to something like root(1600) minus root(1). Nice question.

PrishaAug 5

Yeah, the endpoints basically tell you where it lands. Clever construction.

GuilhermeJul 1

This one took me longer than it should have. Kept trying to find a common denominator before realizing conjugates were the trick. Solid 600-level problem.

KayJul 15

Same here, conjugates didn't jump out at me immediately. Once you see it though it's fast.

What this question tests

What it tests

Your handling of square and higher roots, including simplifying radicals.

Common trap

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.

Find the value of 11+2+12+3+⋯+11599+1600\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \cdots + \frac{1}{\sqrt{1599}+\sqrt{1600}}1​+2​1​+2​+3​1​+⋯+1599​+1600​1​.