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Sn fitness ring
(1) according to the topic, let ⊙Pn radius rn = yn = x2n and area Sn.

∵⊙Pn and ⊙Pn+ 1 are tangent to each other,

∴|PnPn+ 1|=rn+rn+ 1

∴(xn? xn+ 1)2+(yn? yn+ 1)2=yn+yn+ 1,

Square on both sides, arranged (xn? xn+ 1)2=4x2nx2n+ 1

Xn > xn+ 1 > 0,

∴xn-xn+ 1=2xnxn+ 1,

∴ 1xn+ 1? 1xn=2

∴{ 1xn} arithmetic progression, the first term is L, and the tolerance is 2.

∴ 1xn= 1+(n? 1)? 2=2n? 1

∴xn= 12n? 1,

∴rn= 1(2n? 1)2(n∈N*)

(2) the total construction cost of the first n circular paths is t n.

∫Sn =πr2n =πx4n =π(2n? 1)4

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