22.如图,在边长为l的等边△ABC中,圆O1为△ABC的内切圆,圆O2与圆O1

22.如图,在边长为l的等边△ABC中,圆O1为△ABC的内切圆,圆O2与圆O1外切,且与ABBC相切,…,圆On+1与圆On外切,且与ABBC相切,如此无限继续下去.记圆On的面积为annN).

(Ⅰ)证明{an}是等比数列;

(Ⅱ)求

a1+a2+…+an)的值.

答案

22.(Ⅰ)证明:记rn为圆On的半径,则r1=

tan30°=l

=sin30°=.

所以rn=

rn1n≥2),

于是a1r12=

故{an}成等比数列.

 

(Ⅱ)解:因为an=(

n1a1nN),

所以

a1+a2+…+an)==.

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