(Solution) - Let a and b be positive numbers with a -(2025 Original AI-Free Solution)

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Let a and b be positive numbers with a > b. Let a1 be their arithmetic mean and b1 their geometric mean:

Repeat this process so that, in general

(a) Use mathematical induction to show that
an > an+1 > bn+1 > bn
(b) Deduce that both {an} and {bn} are convergent.
(c) Show that lim n?( an = lim n?( bn. Gauss called the common value of these limits the arithmetic-geometric mean of the numbers a and b.