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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8235
GMAT 1: 760 Q51 V42 GPA: 3.82
What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 46% (01:49) correct 54% (02:02) wrong based on 35 sessions

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[GMAT math practice question]

What is the simplified expression for $$\frac{1}{(√1+√2)} + \frac{1}{(√2+√3)} + \frac{1}{(√3+√4)} + ... + \frac{1}{(√99+√100)}?$$

$$A. 8$$
$$B. 9$$
$$C. 10$$
$$D. 11$$
$$E. 12$$

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What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4  [#permalink]

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MathRevolution wrote:
[GMAT math practice question]

What is the simplified expression for $$\frac{1}{(√1+√2)} + \frac{1}{(√2+√3)} + \frac{1}{(√3+√4)} + ... + \frac{1}{(√99+√100)}?$$

$$A. 8$$
$$B. 9$$
$$C. 10$$
$$D. 11$$
$$E. 12$$

Rationalizing denominators of each of the irrational fractions by respective conjugate pairs, we have
$$\frac{(√2-√1)}{(√2+√1)(√2-√1)} +\frac{(√3-√2)}{(√3+√2)(√3-√2)}+---------+\frac{(√100-√99)}{(√100+√99)(√100-√99)}$$
=$$\frac{(√2-√1)}{(2-1)}+ \frac{(√3-√2)}{(3-2)}+---------+\frac{(√100-√99)}{(100-99)}$$
=√2-√1+√3-√2+................+√100-√99
=-1+10
=9

Ans. (B)
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Joined: 09 Apr 2018
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Re: What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4  [#permalink]

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Ans. B

Rationalize the denominator
End up with -9/-1 = 9

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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8235
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4  [#permalink]

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=>

$$\frac{1}{(√1+√2)} + \frac{1}{(√2+√3)} + \frac{1}{(√3+√4)} + ... + \frac{1}{(√99+√100)}$$

$$=\frac{1}{(√2+√1)} + \frac{1}{(√3+√2)} + \frac{1}{(√4+√3)} + ... + \frac{1}{(√100+√99)}$$

$$= \frac{(√2-√1)}{{(√2+√1)(√2-√1)}} + \frac{(√3-√2)}{{(√3+√2)(√3-√2)}} + \frac{(√4-√3)}{{(√4+√3)(√4-√3)}} + ... + \frac{(√100-√99)}{{(√100+√99)(√100-√99)}}$$

$$= (√2-√1) + (√3-√2) + (√4-√3) + ... + (√100-√99)$$

$$= (-√1+√2) + (-√2+√3) + (-√3+√4) + ... + (-√99+√100)$$

$$= -1+10$$

$$= 9$$

_________________ Re: What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4   [#permalink] 07 Apr 2019, 18:43
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# What is the simplified expression for 1/(√1+√2) + 1/(√2+√3) + 1/(√3+√4  