Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 18 Jul 2019, 06:21 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an

Author Message
TAGS:

### Hide Tags

Manager  Joined: 16 Apr 2010
Posts: 193
If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

3
32 00:00

Difficulty:   35% (medium)

Question Stats: 64% (01:15) correct 36% (01:33) wrong based on 491 sessions

### HideShow timer Statistics If the expression $$x=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$$ extends to an infinite number of roots and converges to a positive number x, what is x?

A. $$\sqrt 3$$

B. 2

C. $$1+\sqrt 2$$

D. $$1+\sqrt 3$$

E. $$2*\sqrt 3$$

Originally posted by jakolik on 06 Aug 2010, 01:38.
Last edited by Bunuel on 13 Mar 2019, 04:00, edited 3 times in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 56251
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

9
11
jakolik wrote:
If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an infinite number of roots and converges to a positive number x, what is x?

A- Sqrt(3)
B- 2
C- 1+sqrt(2)
D- 1+sqrt(3)
E- 2*sqrt(3)

$$x=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$$ --> $$x=\sqrt{2+({\sqrt{2+\sqrt{2+\sqrt{2+...})}}}}$$, as the expression under square root extends infinitely then expression in brackets would equal to $$x$$ itself so we can rewrite given expression as $$x=\sqrt{2+x}$$. Square both sides $$x^2=2+x$$ --> $$x=2$$ or $$x=-1$$. As given that $$x>1$$ then only one solution is valid: $$x=2$$.

_________________
##### General Discussion
GMAT Tutor G
Joined: 24 Jun 2008
Posts: 1726
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

5
2
jakolik wrote:
If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an infinite number of roots and converges to a positive number x, what is x?

A- Sqrt(3)
B- 2
C- 1+sqrt(2)
D- 1+sqrt(3)
E- 2*sqrt(3)

This question is out of scope for the GMAT, but there's an interesting trick to questions like this. We know that:

x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...)

Now, notice that the part I've highlighted in red is actually equal to x itself. So we can replace it with x, to get the much simpler equation:

x = sqrt(2 + x)
x^2 = 2 + x
x^2 - x - 2 = 0
(x - 2)(x + 1) = 0
x = 2 or -1

and since x cannot be negative, x = 2. You won't see anything like this on the GMAT though, so it's for interest only.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
Manager  Joined: 06 Apr 2010
Posts: 51
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

That's a clever solution, putting x into the equation itself. Sort of recursive, and very elegant.
These type of problems give me a hard time.
_________________
If you liked my post, please consider thanking me with Kudos! I really appreciate it! Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4487
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

7
1
mun23 wrote:
The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+.....extends to an infinite number of roots.Which of the following choices most closely approximates the value of this expression?

(A)\sqrt{3}
(B)2
(C)1+\sqrt{2}
(D)1+\sqrt{3}
(E)2\sqrt{3}

Dear mun23,
The trick of this question is to give the entire expression a name --- I am going to call it S, and then I am going to square it.
Attachment: nested roots.JPG [ 18.63 KiB | Viewed 157637 times ]

Square the expression produces 2 plus a copy of itself --- that's why we can replace it on the other side with S, and then solve for S algebraically:
S^2 = 2 + S
S^2 - S - 2 = 0
(S - 2)(S + 1) = 1
S = 2 or S = -1
The negative root makes no sense in this context, so S = 2, and the answer = B

Does all this make sense?
Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Manager  Status: struggling with GMAT
Joined: 06 Dec 2012
Posts: 119
Concentration: Accounting
GMAT Date: 04-06-2013
GPA: 3.65
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

Hi mikemcgarry
I am not understanding why the entire expression is given a named?
Magoosh GMAT Instructor G
Joined: 28 Dec 2011
Posts: 4487
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

1
mun23 wrote:
Hi mikemcgarry
I am not understanding why the entire expression is given a named?

Dear mun23,
This is a standard trick in mathematics. We give the entire expression a name, the name S, a variable, because that allows us to manipulate it algebraically. We want to know the value of the entire expression, so we set the entire expression equal to a variable, then ultimately all we have to do is solve for the value of this variable. Because the variable equals the whole expression, when we know the value of the variable, we also know the value of the whole expression.

This is an extension of the fundamental power of algebra --- when we assign a variable to any unknown quantity, then the whole panoply of algebraic techniques comes to bear on the problem.

Keep in mind that material like this ---- infinitely recursive expressions --- is exceedingly unlike to appear on the GMAT. I have never seen anything like this. If it did appear at all, it would only appear to someone getting virtually everything else correct on the Quant section. Folks in the Q < 45 range will NEVER see a question about this stuff, and even folks in the high 50s would only see it less than 1% of the time.

Does all this make sense?

Mike _________________
Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
Math Expert V
Joined: 02 Sep 2009
Posts: 56251
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

mun23 wrote:
The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+.....extends to an infinite number of roots.Which of the following choices most closely approximates the value of this expression?

(A)\sqrt{3}
(B)2
(C)1+\sqrt{2}
(D)1+\sqrt{3}
(E)2\sqrt{3}

I am finding this math quite difficult for me.plz need details explanation...........

Merging similar topics. Please refer to the solutions above.

Similar questions to practice:
tough-and-tricky-exponents-and-roots-questions-125956-40.html#p1029228
find-the-value-of-a-given-a-3-3-3-3-3-inf-138049.html
find-the-value-of-x-75403.html

Hope it helps.
_________________
Manager  Joined: 26 Sep 2013
Posts: 190
Concentration: Finance, Economics
GMAT 1: 670 Q39 V41 GMAT 2: 730 Q49 V41 Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

Bunuel wrote:
jakolik wrote:
If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an infinite number of roots and converges to a positive number x, what is x?

A- Sqrt(3)
B- 2
C- 1+sqrt(2)
D- 1+sqrt(3)
E- 2*sqrt(3)

$$x=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$$ --> $$x=\sqrt{2+({\sqrt{2+\sqrt{2+\sqrt{2+...})}}}}$$, as the expression under square root extends infinitely then expression in brackets would equal to $$x$$ itself so we can rewrite given expression as $$x=\sqrt{2+x}$$. Square both sides $$x^2=2+x$$ --> $$x=2$$ or $$x=-1$$. As given that $$x>1$$ then only one solution is valid: $$x=2$$.

Shouldn't the answer to this be infinity....I have been looking at this one for about 45 minutes, and I can't figure it out. We start with 2, and then add to that $$\sqrt{2}$$, which is about 1.4, then we add to that the square root of the square root of 2, or about 1.18, and then add the square root of the square root of the square root of 2, which is 1.09. The numbers CAN NOT ever be below 1. Just taking it out through 10 cycles, the total is almost 11. And this is an infinite sequence, so the answer is whatever the square root of infinity is. I've looked at the solutions above and they don't make sense to me, at all. The way the problem is written, the answer CANT be any of the options listed
Math Expert V
Joined: 02 Sep 2009
Posts: 56251
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

AccipiterQ wrote:
Bunuel wrote:
jakolik wrote:
If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an infinite number of roots and converges to a positive number x, what is x?

A- Sqrt(3)
B- 2
C- 1+sqrt(2)
D- 1+sqrt(3)
E- 2*sqrt(3)

$$x=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$$ --> $$x=\sqrt{2+({\sqrt{2+\sqrt{2+\sqrt{2+...})}}}}$$, as the expression under square root extends infinitely then expression in brackets would equal to $$x$$ itself so we can rewrite given expression as $$x=\sqrt{2+x}$$. Square both sides $$x^2=2+x$$ --> $$x=2$$ or $$x=-1$$. As given that $$x>1$$ then only one solution is valid: $$x=2$$.

Shouldn't the answer to this be infinity....I have been looking at this one for about 45 minutes, and I can't figure it out. We start with 2, and then add to that $$\sqrt{2}$$, which is about 1.4, then we add to that the square root of the square root of 2, or about 1.18, and then add the square root of the square root of the square root of 2, which is 1.09. The numbers CAN NOT ever be below 1. Just taking it out through 10 cycles, the total is almost 11. And this is an infinite sequence, so the answer is whatever the square root of infinity is. I've looked at the solutions above and they don't make sense to me, at all. The way the problem is written, the answer CANT be any of the options listed

Consider the examples below:
$$\sqrt{2}\approx{1.4}$$;

$$\sqrt{2+\sqrt{2}}\approx{1.85}$$;

$$\sqrt{2+{\sqrt{2+\sqrt{2}}}}\approx{1.96}$$;

$$\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2}}}}}\approx{1.99}$$;

$$\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}\approx{1.998}$$;

$$\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}}}\approx{1.9994}$$.

...

As you can see the result approaches to 2 by decreasing pace. If we extend that to an infinite number of roots the result will be exactly 2.
_________________
Director  S
Joined: 17 Dec 2012
Posts: 630
Location: India
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

jakolik wrote:
If the expression $$x=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$$ extends to an infinite number of roots and converges to a positive number x, what is x?

A. $$\sqrt 3$$

B. 2

C. $$1+\sqrt 2$$

D. $$1+\sqrt 3$$

E. $$2*\sqrt 3$$

Main Idea: Identifying that the problem can be solved by expressing what is given, by an equation. Since there is only one equation possible and if LHS is x , the RHS should also contain x.

Details: We see the given info can be expressed as x=sqrt(2+x). So we can now solve for x . X turns out to be 2 and -1. We reject the negative value and take x=2

Hence B.
_________________
Srinivasan Vaidyaraman
Sravna Test Prep
http://www.sravnatestprep.com

Holistic and Systematic Approach
Non-Human User Joined: 09 Sep 2013
Posts: 11693
Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an   [#permalink] 22 Mar 2019, 11:41
Display posts from previous: Sort by

# If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends to an  