Find the value of x

Question

Find the value of x

x= \sqrt {20+\sqrt {20+\sqrt {20}}}

1. 20
2. 5
3. 2
4. 8

Bunuel · Accepted Answer

Find the value of x x= \sqrt{20+\sqrt{20+\sqrt{20}}} 1. 20 2. 5 3. 2 4. 8 Question should be what is the approximate value of x . Obviously answer choice C (2) is out as \sqrt{20+some \ #}>4 . Now, 4<\sqrt{20}<5 : x= \sqrt{20+\sqrt{20+\sqrt{20}}}= \sqrt{20+\sqrt{20+(# \ less \ than \ 5)}}= \sqrt{20+\sqrt{# \ less \ than \ 25}}= \sqrt{20+(# \ less \ than \ 5)}= =\sqrt{# \ less \ than \ 25}=# \ less \ than \ 5\approx{5} . Answer: B. Next, exactly 5 to be the correct answer question should be: If the expression x=\sqrt{20+{\sqrt{20+\sqrt{20+\sqrt{20+...}}}}} extends to an infinite number of roots and converges to a positive number x, what is x? x=\sqrt{20+{\sqrt{20+\sqrt{20+\sqrt{20+...}}}}} --> x=\sqrt{20+({\sqrt{20+\sqrt{20+\sqrt{20+...})}}}} , 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{20+x} . Square both sides x^2=20+x --> x=5 or x=-4 . As given that x>0 then only one solution is valid: x=5 . Hope it helps.

wcgc · Answer

Since the answer choices are dissimilar, we can estimate the answer choice here. The  \sqrt {20}  is somewhere between 4 and 5. Suppose it's 5, then we'll get
 \sqrt {20+\sqrt {20+5} = \sqrt {20+5} = 5

FN · Answer

agree with 5..

i estimated it to be 5..

now if they had a 4 in the ans choices..that would have been tough..

x2suresh · Answer

even if you have 4.. its not tough.. sqrt(20) clearly.. >4

I agree if answer choice has options like 4.9 or 4.8...

x2suresh · Answer

actual value of x= 4.994690378 ~5

only way to do is process of elimination.

GMAT TIGER · Answer

1:  x= \sqrt{20+\sqrt{20+\sqrt{20}}} 
 x= \sqrt{20+\sqrt{20+4.47}} 
 x= \sqrt{20+\sqrt{24.47}} 
 x= \sqrt{20+4.95} 
 x= \sqrt{24.95} 
 x= 4.995 = approx. 5.00

2: From third sqrt: sqrt 20 = 4 and a fraction of 1.
From second sqrt: sqrt (20+4.00 = 24) = 4 and a fraction of 1.
From first sqrt: sqrt (20+4.00 = 24) = 4 and a fraction of 1. which is definitely close to none other than 5.

3: Using POE:

A: it cannot be 20 cuz for 20, the value under root must be 400, which is impossible. so A is ruled out.
B: It could be 5 as done above in method 2.
3. 2 is also not possible because even if we consider first 20 under root, the value must not be smaller than 4.
4. 8 is not possible because for 8, the value under root must be 64. Even if we add up all three 20s, the sum would not be more than 60. so it is also not possible. So left with 5.

So B make sense.

mrsmarthi · Answer

If so, shouldn't the question be "What is the approximate value of x?" Just curious :wink:

neomax85 · Answer

x=root of 20+x
x^2=20+x
x^2-x=20
solving we get x=5,-4

Aztec · Answer

boiled it down to roughly 5.5ish. 8 is too high. must be 5.

nonameee · Answer

Bunuel, would you be so kind and look at this question. Is there any other way to solve it rather than elimination? Can you describe elimination in greater detail? Thank you.

nonameee · Answer

Bunuel, thank you very much.

Bunuel · Answer

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 if-the-expression-x-sqrt-2-sqrt-2-sqrt-2-sqrt-2-extends-98647.html Hope it helps.

PareshGmat · Answer

x= \sqrt {20+\sqrt {20+\sqrt {20}}}

Converting the square root sign to power

x = (20 + (20 + (20)^{\frac{1}{2}})^{\frac{1}{2}})^{\frac{1}{2}}

Squaring both sides

x^2 = 20 + (20 + (20)^{\frac{1}{2}})^{\frac{1}{2}}

Now start looking at the OA

Value of  x^2  has to be around 25, but way less than 64

Answer = 5 = B

rhine29388 · Answer

why is it that only 4 options given in the question?
I think that this is not a gmat type question
as for explanation,I agree with bunuel that it needs to be asking approximate value of x
Correct Ans - B

LogicGuru1 · Answer

Fortunately there is a super short cut to solve this question (that shortcut is applicable to this question atleast

) This whole expression can be seen as x= \sqrt {20+\sqrt {some number} . Square root of 20 is 4.4 approx and the rest of the nested square root will be a small decimal value because they are square root inside square root so the answer will be 4.4 + 0.some value Look out for a nearest answer greater than 4.4 It's 5 in the given question SO the answer is B If you get this kind of questions in GMAT , thank your stars because you can save tremendous amount of time and add sure marks to your score.

MikeScarn · Answer

Am I wrong to convert this to  20^\frac{1}{2}  +  20^\frac{1}{4}  +  20^\frac{1}{8}  ?

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Find the value of x

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