If an=7+7+7+⋯having n radical signs then by methods of mathematical induction which of the following is true

If $a_{n} = \sqrt{7 + \sqrt{7 + \sqrt{7 + \dots}}}$ having n radical signs then by methods of mathematical induction which of the following is true

We have $a_{1} = \sqrt{7} < 4$ Assume $a_{k} < 4$ for

some natural number $k$

We have

$\begin{array}{l} a_{k + 1} = \sqrt{7 + a_{k}} < \sqrt{7 + 4} < 4 \\ ∴ a_{n} < 4 \forall n \geq 1 . \end{array}$

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