Question

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

• A. $a_n<7\forall n\ge 1$
• B. $a_n>7\forall n\ge 1$
• C. $a_n<4\forall n\ge 1$
• D. $a_n<3\forall n\ge 1$

Answer 1

The correct option is A

$a_n<7\forall n\ge 1$

$a_1=\sqrt{7}<7$. Let a_m < 7.
Then $a_{m+1}=\sqrt{7+a_m}$
$\Rightarrow a_{m+1}^2=7+a_m<7+7<14\Rightarrow a_{m+1}<\sqrt{14}<7$
So $a_m<7\forall n\ge 1$