If z^2= 2+ 2+ 2+ 2+..., where the given expre… | GMAT Sprint

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PS · Problem Solving

705-805

Roots

If $z^2=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$ , where the given expression extends till infinity, which of the following statements must be true ?
I. Two values are possible for z
II. 4 - z^2 = 2
III. z^8 = 16

Answer Choices

Correct answer marked below

I only

II only

III only

I, II and III

Correct

None of the above

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Explanation

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Official Explanation

Official

Given:

z^2=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}

.

Since the expression under the square root extends infinitely, the part inside the outer square root after the first 2 is the same as the original expression. So we can write:

z^2=\sqrt{2+z^2}

.

Square both sides:

z^4=2+z^2

.

Rearrange:

z^4 - z^2 - 2 = 0

.

Let

x = z^2

, then:

x^2 - x - 2 = 0

.

Solve:

(x - 2)(x + 1) = 0

.…

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What this question tests

What it tests

Your handling of square and higher roots, including simplifying radicals.

Common trap

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.

If z2=2+2+2+2+...z^2=\sqrt{2+{\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}z2=2+2+2+2+...​​​​, where the given expression extends till infinity, which of the following statements must be true ?I. Two values are possible for zII. 4 - z^2 = 2III. z^8 = 16