Maths等比/等差數列一問 3

ai) B1B2C2D2周界:A1B1C1D1周界
=4(B1B2):4(A1B1)
=(B1B2):(A1B1)
=(1/2)(A1B1):(A1B1)
=1:2
=0.5
ii)B1B2C2D2面積:A1B1C1D1面積
=((B1B2):(A1B1))^2
=(1:2)^2
=1^2:2^2
=1:4
=0.25

bi)設n為正方形總數
18cm為A1B1的長度
等比=1/2
周界總和
=4*(18(1-(1/2)^n)/(1-1/2)
當n趨向無限, (1/2)^n趨向0
=4*(18(1-0)/(1-1/2))
=144cm
ii)設n為正方形總數
18^2=324cm^2為A1B1的面積
等比=1/4
面積總和
=324(1-(1/4)^n)/(1-1/4)
當n趨向無限, (1/4)^n趨向0
=324(1-0)/(1-1/4)
=432cm^2

20(a)(i) A1B1 = 18 cm. B1B2 = 9 cm正方形A1B1C1D1的周界 = 72 cm。面積 = 324 cm^2正方形B1B2C2D2的周界 = 36 cm。面積 = 81 cm^2正方形B1B2C2D2的周界/正方形A1B1C1D1的周界 = 1/2(ii) 正方形B1B2C2D2的面積/正方形A1B1C1D1的面積 = 1/4(b)(i) 首項 = 72﹐公比 = 1/2周界總和 = a/(1 - r) = 72/(1 - 1/2) = 144 cm(ii) 首項 = 324﹐公比 = 1/4面積總和 = a/(1 - r) = 324/(1 - 1/4) = 432 cm^2