Question

If $a_n=\sqrt{7+\sqrt{7+\sqrt{7+.....}}}$ having $n$ radical signs, then by method of mathematical induction which of the following is true?

• A. $a_n>6,\forall n>1$
• B. $a_n>3,\forall n>1$
• C. $a_n>4,\forall n>1$
• D. $a_n<2,\forall n>1$

Answer 1

The correct option is B $a_n>3,\forall n>1$

$a_n=\sqrt{7+\sqrt{7+\sqrt{7}+.....}}$

$\Rightarrow a_n=\sqrt{7+a_n}\Rightarrow a_n^2-a_n-7=0$

$\therefore a_n=\frac{1\pm \sqrt{1+28}}{2}=\frac{1\pm \sqrt{29}}{2}$

But $a_n>0\therefore a_n=\frac{1+\sqrt{29}}{2}>3$