If $a_n=\sqrt{7+\sqrt{7+\sqrt{7+\ldots n}}}$ times, then which one of the following is true?

If $a_n=\sqrt{7+\sqrt{7+\sqrt{7+\ldots n}}}$ times, then which one of the following is true?
  1. $a_n>7, \forall n \geq 1$
  2. $a_n>3, \forall n \geq 1$
  3. $a_n < 4, \forall n \geq 1$
  4. $a_n < 3, \forall n \geq 1$

Solution

On limiting case $a_n < a_{\infty}$ Now, $\quad a_{\infty}=\sqrt{7+a_{\infty}}$ $ \Rightarrow a_{\infty}^2-a_{\infty}-7=0 \Rightarrow a_{\infty}=\frac{1+\sqrt{29}}{2} \quad\left[\because a_{\infty}>0\right] $ So, $\quad a_n < \frac{1+\sqrt{29}}{2}, \forall n \geq 1$ $ \therefore \quad a_n < 4, \forall n \geq 1 . $

Asked in: AP EAMCET 2018 (22 Apr Shift 2)

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