# Find the value of $\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } }$

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Find the value of $\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } }$
1. 3
2. 5
3. 6
4. 4

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Correct Answer - Option 3 : 6

Given:

$\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } }$

Calculation:

Let 'a' be the value of $\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } }$

⇒ a = $\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } }$

Squaring both sides, we get

⇒ a2 = 30 + $\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } }$

⇒ a2 = 30 + a

⇒ a2 - a - 30 = 0

⇒ a2 - 6a + 5a - 30 = 0

⇒ (a - 6)(a + 5) = 0

⇒ a = 6, -5

a = - 5 is not possible

∴ The value of $\sqrt {30 + \sqrt {30 + \sqrt {30 + \sqrt {30 + ...\,\,...\,\,...\,\,...\,\,...\,\,.\,\infty } } } }$ is 6.