Some simple sequences

Calculus Level 5

The three sequences, \( \{ a_n \}_{n=1}^{\infty } , \{ b_n \}_{n=1}^{\infty } \) and \( \{ c_n \}_{n=1}^{\infty } \), satisfy the following:

  1. \( \displaystyle a_n \leq b_n \leq c_n \quad \forall n \in \mathbb{ N} \)

  2. \( \displaystyle a_n + b_n + c_n = p(n) \) ; which is a polynomial.

  3. \( \displaystyle a_n b_n c_n \) is a non zero constant \( \forall n \in \mathbb{ N}\).

  4. \( \displaystyle a_n + \frac{1}{a_{n+1}} + b_n + \frac{1}{b_{n+1}} + c_n + \frac{1}{c_{n+1}} = 0 \quad \forall n \in \mathbb{ N}\).

  5. \( \displaystyle {a_n}^3 + {b_n}^3 + {c_n}^3 = 8n^3 + 6n +1 \).

Evaluate: \( \displaystyle \lim_{n \to \infty } ( 38n a_n ) \)

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