First slide

Introduction to P.M.I

Question

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

Moderate

A

$a_{n} > 7 \forall n \geq 1$
B

$a_{n} > 3 \forall n \geq 1$
C

$a_{n} < 4 \forall n \geq 1$
D

$a_{n} < 3 \forall n \geq 1$

Solution

We have $a_{1} = \sqrt{7} < 4$ Assume $a_{k} < 4$ ,for some natural number k.
We have $a_{k + 1} = \sqrt{7 + a_{k}} < \sqrt{7 + 4} < 4$
$∴ a_{n} < 4 \lor n \geq 1$ .

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