Question

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

Options

• a_n > 7 ∀ n ≥ 1

• a_n < 7 ∀ n ≥ 1

• a_n < 4 ∀ n ≥ 1

• a_n < 3 ∀ n ≥ 1

Answer 1

a_n < 7 ∀ n ≥ 1

Explanation:

For n = 1, a₁ = `sqrt(7) < 7`.

Let a_m < 7.

Then a_m+1 = `sqrt(7 + a_m)`

⇒ a_m+1 = 7 + a_m < 7 + 7 < 14.

⇒ a_m+1 < `sqrt(14) < 7`; So, by the principle of mathematical induction a_n < 7 ∀ n.