Geometric mean ($G=(x_1\cdots x_n{)}^{1/n}$)

The geometric mean of finitely many nonnegative real numbers is the nonnegative $n$th root of their product. For two positive numbers it is $\sqrt{ab}$. The arithmetic-geometric mean inequality compares it with their arithmetic mean. Unlike an arithmetic average, the geometric mean respects multiplicative scaling and is natural for growth factors.

Starting from $0<a_1\le b_1$, replace the lower endpoint by the geometric mean and the upper endpoint by the arithmetic mean. The arithmetic-geometric mean inequality gives $a_n\le a_{n+1}\le b_{n+1}\le b_n$. The bounded monotone sequence theorem gives limits, and the arithmetic recurrence forces them to coincide. The limiting value is the arithmetic-geometric mean of the initial pair.