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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.

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