Question 81
A real sequence $(a_n)_{n=0}^{\infty}$ is defined recursively by $a_0 = 2$ and the recursion formula
$$
a_n = \begin{cases} a_{n-1}^2 & \text{if } a_{n-1} \lt \sqrt{3} \\ \frac{a_{n-1}^2}{3} & \text{if } a_{n-1} \ge \sqrt{3}. \end{cases}
$$
Another real sequence $(b_n)_{n=1}^{\infty}$ is defined in terms of the first by the formula
$$
b_n = \begin{cases} 0 & \text{if } a_{n-1} \lt \sqrt{3} \\ \frac{1}{2^n} & \text{if } a_{n-1} \ge \sqrt{3}, \end{cases}
$$
valid for each $n \ge 1$. Prove that
$$
b_1 + b_2 + \dots + b_{2020} \lt \frac{2}{3}.
$$
$$
a_n = \begin{cases} a_{n-1}^2 & \text{if } a_{n-1} \lt \sqrt{3} \\ \frac{a_{n-1}^2}{3} & \text{if } a_{n-1} \ge \sqrt{3}. \end{cases}
$$
Another real sequence $(b_n)_{n=1}^{\infty}$ is defined in terms of the first by the formula
$$
b_n = \begin{cases} 0 & \text{if } a_{n-1} \lt \sqrt{3} \\ \frac{1}{2^n} & \text{if } a_{n-1} \ge \sqrt{3}, \end{cases}
$$
valid for each $n \ge 1$. Prove that
$$
b_1 + b_2 + \dots + b_{2020} \lt \frac{2}{3}.
$$