Find the value of \(\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+...}}}}}=?\)

Question

Question

This question was previously asked in

NPCIL Assistant Grade-1 (Advance) Official Paper-II (Held On: 08 Sept, 2018)

Answer 1

Given:

Find the value of √(6 + √(6 + √(6 + √(6 + ...)))) = ?

Formula used:

Let x = √(6 + √(6 + √(6 + ...)))

x² = 6 + x

Calculation:

x² = 6 + x

⇒ x² - x - 6 = 0

Solving the quadratic equation:

x = [1 ± √(1² - 4×1×(-6))] / (2×1)

⇒ x = [1 ± √(1 + 24)] / 2

⇒ x = [1 ± √25] / 2

⇒ x = [1 ± 5] / 2

⇒ x = 6 / 2 or x = -4 / 2

⇒ x = 3 or x = -2

Since x must be positive, we choose x = 3.

Hence, the value of the given expression is 3.

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