Problem 91E

91. Let $a$ and $b$ be positive numbers with $a>b$. Let $a_{1}$ be their arithmetic mean and $b_{1}$ their geometric mean:

$$a_{1}=\frac{a+b}{2} \quad b_{1}=\sqrt{a b}$$

Repeat this process so that, in general,

$$a_{n+1}=\frac{a_{n}+b_{n}}{2} \quad b_{n+1}=\sqrt{a_{n} b_{n}}$$

(a) Use mathematical induction to show that

$$a_{n}>a_{n+1}>b_{n+1}>b_{n}$$

(b) Deduce that both $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are convergent.

(c) Show that $\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .$ Gauss called the common value of these limits the arithmetic-geometric mean of the numbers $a$ and $b$.

Step-by-Step Solution