急有關Geometric Sequences!

[匿名]

2011-09-14 4:33 am

1. A ball falls from a height of 10m to the ground, then it rebounds to a height of 6m. The ball falls again and then rebounds to a height which is 0.6 times that of the preceding rebound(i.e. 6m). If the ball continuously falls and rebounds in this way, find the total vertical distance travelled by the ball just before the 5th rebound.2 .The radius of circle C1 is 1cm, the radius of circle C2 is 2 times that of C1, radius of circle C3 is 2times that of circle C2, and the same method is used to construct a set of circles C1,C2,C3…(a) Express the area of C1,C2 and C3 in terms of π(b) find the total area of the circles C1,C2,C3….,C7,and express the answer in terms of π. 3. A long string is cut into 6 pieces, and the lengths of these 6 pieces of string form a geometric sequence. The lengths of the first piece and the last piece are 64cm and 2cm respectively.(a) What is the length of the original string?(b) Suppose the original string is cut into 3 pieces. The lengths of these 3 pieces of string form another geometric sequence. If the length of the

回答 (1)

fooks

2011-09-14 9:36 pm

✔ 最佳答案

1.
Total vertical distance travelled
= [10 + 2*10*(0.6) + 2*10*(0.6)² + 2*10*(0.6)³] m
= {10 + 2*10*(0.6)[1 + (0.6) + (0.6)²]} m
= {10 + 12[1 - (0.6)³]/(1 - 0.6)} m
= 33.52 m

2 .
(a)
Area of C1
= π(1)² cm²
= π cm²

Area of C2
= π(1 * 2)²
= 4π cm²

Area of C3
= π (1 * 2 * 2)²
= 16π cm²

(b)
Total area of C1, C2, C3, C4, C5, C6, C7
= [π + π(4) + π(4)² + π(4)³ + π(4)⁴ + π(4)⁵ + π(4)⁶] cm²
= π[(4)⁷ - 1] / (4 - 1) cm²
= 5461π cm²

3. A long string is cut into 6 pieces, and the lengths of these 6 piecesof string form a geometric sequence. The lengths of the first piece and thelast piece are 64cm and 2cm respectively.
(a) What is the length of theoriginal string?
(b) Suppose the original stringis cut into 3 pieces. The lengths of these 3 pieces of string form anothergeometric sequence. If the length of the

(a)
The first term, T(1) = a = 64
The common ratio = r

T(6):
64 * r⁵ = 2
r⁵ = 1/32
r = ⁵√(1/32)
r = 1/2

Length of the original string
= [64 + 64(1/2) + 64(1/2)² + 64(1/2)³ + 64(1/2)⁴ + 64(1/2)⁵] cm
= 64[1 - (1/2)⁶] / [1 - (1/2)] cm
= 126 cm

(b)
The question is incomplete, and something has been missed.

參考: Tsui

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