Question 1
For a real number $x$, let $[x]$ denote the greatest integer not exceeding $x$. Define $a_n=\dfrac1{4^{[-\log_4 n]}}$ for $n\ge1$, and put
$$b_n=\frac1{n^2}\left(\sum_{k=1}^na_k-\frac1{a_1+a_2}\right).$$
(a) Find a real-coefficient polynomial $P(x)$ such that $b_n=P(a_n/n)$ for every $n\ge1$.
(b) Prove that there is a strictly increasing sequence of positive integers $(n_k)$ such that $\displaystyle\lim_{k\to\infty}b_{n_k}=2024/2025$.
$$b_n=\frac1{n^2}\left(\sum_{k=1}^na_k-\frac1{a_1+a_2}\right).$$
(a) Find a real-coefficient polynomial $P(x)$ such that $b_n=P(a_n/n)$ for every $n\ge1$.
(b) Prove that there is a strictly increasing sequence of positive integers $(n_k)$ such that $\displaystyle\lim_{k\to\infty}b_{n_k}=2024/2025$.