A) \[{{a}_{n}}>7\forall n\ge 1\]
B) \[{{a}_{n}}<7\forall n\ge 1\]
C) \[{{a}_{n}}<4\forall n\ge 1\]
D) \[{{a}_{n}}<3\forall n\ge 1\]
Correct Answer: B
Solution :
| \[{{a}_{1}}=\sqrt{7}<7.\]Let \[{{a}_{m}}<7\] |
| Then\[{{a}_{m+1}}=\sqrt{7+{{a}_{m}}}\Rightarrow {{a}^{2}}_{m+1}=7+{{a}_{m}}\] |
| \[<7+7<14.\] |
| \[\Rightarrow \]\[{{a}_{m+1}}<\sqrt{14}<7;\]so by the principle of mathematical induction \[{{a}_{m+1}}<\sqrt{14}\forall n.\] |
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