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Let $a_n$ denote the nested radical with $n$ square-root signs, defined by $a_n=\sqrt{7+\sqrt{7+\sqrt{7+\cdots}}}$. Using mathematical induction, which statement is true for all $n\ge 1$?

1. $a_n>7$ for every $n\ge 1$
2. $a_n<7$ for every $n\ge 1$
3. $a_n<4$ for every $n\ge 1$
4. $a_n<3$ for every $n\ge 1$

Correct answer: $a_n<7$ for every $n\ge 1$

Solution

The nested radical terms are positive and remain bounded. Since $a_{n+1}=\sqrt{7+a_n}$, if $a_n<7$ then $a_{n+1}<\sqrt{14}<7$, so the property is preserved by induction. Thus $a_n<7$ for all $n\ge 1$.

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