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If $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } } $$ having $$n$$ radical signs then by methods of mathematical induction which is true

JEE · Math · previous-year question

  1. A.$${a_n} > 7\,\,\forall \,\,n \ge 1$$
  2. B.$${a_n} < 7\,\,\forall \,\,n \ge 1$$
  3. C.$${a_n} < 4\,\,\forall \,\,n \ge 1$$
  4. D.$${a_n} > 3\,\,\forall \,\,n \ge 1$$correct

Answer

D. $${a_n} > 3\,\,\forall \,\,n \ge 1$$

Explanation

Given $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } } $$ $$\therefore$$ $${a_n} = \sqrt {7 + {a_n}} $$ $$ \Rightarrow $$ $$a_n^2 = 7 + {a_n}$$ $$ \Rightarrow $$ $$a_n^2 - {a_n} - 7 = 0$$ $$ \Rightarrow {a_n} = {{1 \pm \sqrt {1 - 4 \times 1 \times - 7} } \over 2}$$ $$ \Rightarrow {a_n} = {{1 \pm \sqrt {29} } \over 2}$$ As $${a_n}$$ > 0, $$\therefore$$ $${a_n} = {{1 + \sqrt {29} } \over 2}$$ = 3.19 So $${a_n} > 3\,\,\forall \,\,n \ge 1$$

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