Let a and b be positive numbers with a

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} $ $ b_1 = \sqrt {ab} $ Repeat this process so that, in general, $ a^{n + 1} = \frac {a_n + b_n}{2} $ $ 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 $ \{ a_n \} $ and $ \{ b_n \} $ are convergent. (c) Show that $ \lim_{n \to\infty} a_n = \lim_{n \to \infty} b_n $. Gauss called the common value of these limits the arithmetic-geometric mean of the numbers $ a $ and $ b. $

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} $
$ b_1 = \sqrt {ab} $
Repeat this process so that, in general,
$ a^{n + 1} = \frac {a_n + b_n}{2} $
$ 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 $ \{ a_n \} $ and $ \{ b_n \} $ are convergent.
(c) Show that $ \lim_{n \to\infty} a_n = \lim_{n \to \infty} b_n $. Gauss called the common value of these limits the arithmetic-geometric mean of the numbers $ a $ and $ b. $

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