If the expression $\backslash (\backslash sqrt\{2+\backslash sqrt\{2+\backslash sqrt\{2+\backslash sqrt\{2+....\}\}\}\}\backslash )$ extends to an infinite number of roots GMAT Problem Solving

Question: If the expression \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+....}}}}\) extends to an infinite number of roots and converges to a positive number x, what is x?

1. \(\sqrt3\)
2. 2
3. \(1+\sqrt{2}\)
4. \(1+\sqrt{3}\)
5. \(2*\sqrt{3}\)

‘If the expression \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+....}}}}\) extends to an infinite number of roots and converges to a positive number x, what is x?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “Official Guide for GMAT Reviews”. To solve GMAT Problem Solving examiners measure how well the candidates make analytical and logical approaches to solve numerical problems. In this section, candidates have to evaluate and interpret data from given graphical representation. In this section, mostly one finds out mathematical questions. Five answer choices are given for each GMAT Problem solving question.

Solution and Explanation:

Approach Solution 1:

There is only one approach to solve this problem.

x=\(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+....}}}}\)

Let \(\sqrt{2+\sqrt{2+\sqrt{2+}}}\) be x.

So we can replace the part with x to make the equation simpler to solve.

x = \(\sqrt{(2+x)}\)

that implies \(x^2\)= 2 + x

that implies\(x^2\) - x - 2 = 0

that implies (x - 2)(x + 1) = 0
that implies x = 2 or -1
and since x cannot be negative, x = 2.
Therefore, the required value of x is 2. Hence option B is the correct answer.

Correct Answer: B

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