Question
If $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + ......} } } $$ having $$n$$ radical signs then by methods of mathematical induction which is true
A.
$${a_n} > 7\,\,\forall \,\,n \geqslant 1$$
B.
$${a_n} < 7\,\,\forall \,\,n \geqslant 1$$
C.
$${a_n} < 4\,\,\forall \,\,n \geqslant 1$$
D.
$${a_n} < 3\,\,\forall \,\,n \geqslant 1$$
Answer :
$${a_n} < 7\,\,\forall \,\,n \geqslant 1$$
Solution :
$$\eqalign{
& {a_1} = \sqrt 7 < 7.\,\,{\text{Let }}{a_m} < 7 \cr
& {\text{Then }}{a_{m + 1}} = \sqrt {7 + {a_m}} \cr
& \Rightarrow \,\,{a^2}_{m + 1} = 7 + {a_m} < 7 + 7 < 14. \cr
& \Rightarrow \,\,{a_{m + 1}} < \sqrt {14} < 7; \cr} $$
So by the principle of mathematical induction $${a_n} < 7\,\,\forall \,\,n.$$