87. Given positive numbers aj < b1, define two sequences recursively by an + bn 2 An+1 = Va,bn, bn+1 (a) Show that a, s b, for all n (Figure 14). (b) Show that {an) is increasing and {bn} is decreasing. bn - an (c) Show that b+1 – an+1 3 2 (d) Prove that both {a,} and {b,} converge and have the same limit. This limit, denoted AGM(aj, bị), is called the arithmetic-geometric mean of aj and b1. (e) Estimate AGM(1, /2) to three decimal places. Geometric Arithmetic mean mean AGM(a, b,) FIGURE 14

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