Question:  If x is a positive number and equals to\( \sqrt{6+ \sqrt{6+\sqrt{6+\sqrt{6+}}}}\).... where the given expression extends to an infinite number of roots, then what is the value of x?

1. √6
2. 3
3. 1+√6
4. 2√3
5. 6

Answer: B

Solution and Explanation:

Approach Solution 1:

Given: x > 0,

x=\( \sqrt{6+ \sqrt{6+\sqrt{6+\sqrt{6+}}}}\).... Re-write it as: x = \( \sqrt{6+ \sqrt{6+\sqrt{6+\sqrt{6+}}}}\)...._)

as the expression under the square root extends infinitely then the expression in brackets would equal to x itself and we can safely replace it with x and rewrite the given expression as

x = \(\sqrt{6+x}\)

Square both sides:

\(x^2\) = 6 + x

\(x^2\) - x - 6 = 0

Now, re-arrange and factorize

(x+2)(x−3)=0

so x = −2 or x = 3 but since x>0 then: x=3

B is the correct answer.

Approach Solution 2:

Solve by approximation:

√6 ≈ 2.4495

\(\sqrt{6 +\sqrt{6}}\) ≈ 2.90686

\(\sqrt{6+\sqrt{6+\sqrt{6}}}\) ≈ 2.984436

\( \sqrt{6+ \sqrt{6+\sqrt{6+\sqrt{6}}}}\) ≈ 2.99746

It can be seen that the values are approaching 3.

B is the correct answer.

“If x is a positive number and equals to" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Club tests questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude for reasoning and mathematics. The GMAT Quantitative Test's Club Test Phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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