Question 186
Let $a_1, a_2, \ldots$ be the sequence given by $a_1 = 1$ and $a_{n+1} = \frac{2n+1}{2n+4} a_n$ for $n \ge 1$.
(1)
Prove that the sequence $b_n = a_n^2 \times n(n+1)^2$ is increasing.
(2)
Prove
$$
\left(\frac{1}{n}\right)^{3/2} \le a_n \lt \left(\frac{2}{n}\right)^{3/2}
$$
for all $n$.
(1)
Prove that the sequence $b_n = a_n^2 \times n(n+1)^2$ is increasing.
(2)
Prove
$$
\left(\frac{1}{n}\right)^{3/2} \le a_n \lt \left(\frac{2}{n}\right)^{3/2}
$$
for all $n$.