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.