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

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If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends [#permalink] New post 06 Aug 2010, 01:38
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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)
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Re: Interesting Square Root Problem [#permalink] New post 06 Aug 2010, 01:51
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.
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Re: Interesting Square Root Problem [#permalink] New post 06 Aug 2010, 01:51
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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.

Answer: B.
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Re: Interesting Square Root Problem [#permalink] New post 06 Aug 2010, 11:59
Thanks for the explanation. Its helpful
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Re: Interesting Square Root Problem [#permalink] New post 06 Aug 2010, 22:45
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.
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Sum of an infinite series [#permalink] New post 26 Aug 2012, 12:29
Please let me know if you get the formula.
I have to find the sum of the square root of 2 plus Sq root of sq root of 2 plus.... (I have pasted a pic)...
Attachments

File comment: This series goes on forever. I could draw only three twos, but the no is infinite
Sum of infinite root sequence.png
Sum of infinite root sequence.png [ 7.4 KiB | Viewed 1320 times ]

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Re: If the expression x=sqrt(2+sqrt(2+sqrt(2+sqrt(2+...) extends [#permalink] New post 20 Nov 2012, 19:39
It is the √2 plus 2^1/4 + 2^1/8...

all of those tiny little bits of 2, if they extend to infinity, end up to being another 2 under the radical.

√2+2 = 2.
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The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+..... [#permalink] New post 27 Feb 2013, 11:12
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...........
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Re: The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+..... [#permalink] New post 27 Feb 2013, 13:03
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
nested roots.JPG [ 18.63 KiB | Viewed 441 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 :-)
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Re: The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+..... [#permalink] New post 27 Feb 2013, 13:29
Hi mikemcgarry
I am not understanding why the entire expression is given a named?
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Re: The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+..... [#permalink] New post 27 Feb 2013, 14:38
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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 :-)
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Re: The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+..... [#permalink] New post 27 Feb 2013, 15:18
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.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
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Re: The expression sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+.....   [#permalink] 27 Feb 2013, 15:18
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