# 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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