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

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

1. A

${a}_{n}>7\mathrm{\forall }n\ge 1$

2. B

${a}_{n}>3\mathrm{\forall }n\ge 1$

3. C

${a}_{n}<4\mathrm{\forall }n\ge 1$

4. D

${a}_{n}<3\mathrm{\forall }n\ge 1$

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### Solution:

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

some natural number $k$

We have

$\begin{array}{l}{a}_{k+1}=\sqrt{7+{a}_{k}}<\sqrt{7+4}<4\\ \therefore {a}_{n}<4\mathrm{\forall }n\ge 1.\end{array}$

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