If `a_n = sqrt(7+ sqrt( 7+sqrt7+ ....... )))` having n radical signs then by methods of mathematical induction which is true

Options

A. `a_n gt 7,forall n ge 1`
B. `n_ngt 3,foralln ge 1`
C. `a_nlt 4, forall n ge 1`
D. `a_nlt 3,forall n ge 1`

Explanation

To solve the problem using mathematical induction, we need to analyze the expression given:

**Step 1: Define the expression.**
Let \( a_n = \sqrt{7 + \sqrt{7 + \sqrt{7 + \ldots}}} \) (with \( n \) radical signs).

**Step 2: Base case.**
We start with the base case for \( n = 1 \):
\[
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