Question 172
Let $b \gt a \gt 1$ be real numbers. Define
$$
x_n = 2^n \left( \sqrt[2^n]{b} - \sqrt[2^n]{a} \right).
$$
Prove that the sequence $x_1, x_2, \dots$ is strictly decreasing.
$$
x_n = 2^n \left( \sqrt[2^n]{b} - \sqrt[2^n]{a} \right).
$$
Prove that the sequence $x_1, x_2, \dots$ is strictly decreasing.