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JEE Main 2002

Exam: JEE Main
Year: 2002
Subject: Mathematics
Chapter: Mathematical Induction

JEE Main 2002 · Mathematics · Mathematical Induction

If a_n= 7+ 7+ 7+ . having n radical signs then by methods of...

If

a_n=\sqrt{7+\sqrt{7+\sqrt{7+\ldots .}}}

having

\mathrm{n}

radical signs then by methods of mathematical induciton which is true

Options

A $a_n>7 \forall n \geq 1$
B $\mathrm{a}_{\mathrm{n}}>7 \forall \mathrm{n} \geq 1$
C $a_n < 4 \forall n \geq 1$
D $a_n < 3 \forall \mathrm{n} \geq 1$

Correct answer

B. $\mathrm{a}_{\mathrm{n}}>7 \forall \mathrm{n} \geq 1$

Step-by-step solution

\mathrm{a}_1=\sqrt{7} < 7

. Leta

_{\mathrm{m}} < 7

. Then

\mathrm{a}_{\mathrm{m}}+1=\sqrt{7+\mathrm{a}_{\mathrm{m}}} \Rightarrow \mathrm{a}_{\mathrm{m}+1}^2=7+\mathrm{a}_{\mathrm{m}} < 7+7 < 14

\Rightarrow \mathrm{a}_{\mathrm{m}+1} < \sqrt{14} < 7 ;

So

\mathrm{a}_{\mathrm{n}} < 7 \forall \mathrm{n} \therefore \mathrm{a}_{\mathrm{n}}>3

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