Question 160
Given the sequence $\{a_n\}_{n=1}^{\infty}$ defined by
$$
a_n = \frac{1}{4^{[-\log_4 n]}}
$$
for all positive integers $n$. Put
$$
b_n = \frac{1}{n^2} \left( a_1 + a_2 + \cdots + a_n - \frac{1}{a_1 + a_2} \right), \quad \forall n \in \mathbb{Z}^+
$$
a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P\left(\frac{a_n}{n}\right)$ for all positive integers $n$.
b) Prove that there exists a strictly increasing sequence of positive integers $\{n_k\}_{k=1}^{\infty}$ such that
$$
\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}
$$
$$
a_n = \frac{1}{4^{[-\log_4 n]}}
$$
for all positive integers $n$. Put
$$
b_n = \frac{1}{n^2} \left( a_1 + a_2 + \cdots + a_n - \frac{1}{a_1 + a_2} \right), \quad \forall n \in \mathbb{Z}^+
$$
a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P\left(\frac{a_n}{n}\right)$ for all positive integers $n$.
b) Prove that there exists a strictly increasing sequence of positive integers $\{n_k\}_{k=1}^{\infty}$ such that
$$
\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}
$$