急 F6 Geometric Sequences 02q16

35.
(a)
The total length of the ant's route
= 200 + 100 + 50 + ......

This is the sum to infinite terms of a geometric sequence.
The first term = 200
The common ratio = (1/2)

The total length of the ant's route
= 200 / [1 - (1/2)]
= 400 (units)

(b)
The x coordinate of the final position of the ant
= 100 - 100x(1/2)² + 100x(1/2)⁴ - ......
= 100 - 100x(1/4) + 100x(1/4)² - ......

This is the sum to infinite terms of a geometric sequence.
The first term = 100
The common ratio = -1/4

The x coordinate of the final position of the ant
= 100 / [1 - (-1/4)]
= 80

The y coordinate of the final position of the ant
= 200 - 50 + 50x(1/2)² - 50x(1/2)⁴ + ......
= 200 - (50 - 50x(1/4) + 50x(1/4)² + ......)

50 - 50x(1/4) + 50x(1/4)² + ...... is the sum to infinite terms of a geometric sequence.
The first term = 50
The common ratio = -1/4

The y coordinate of the final position of the ant
= 200 - {50 / [1 - (-1/4)]}
= 160

The coordinates of the final position of the ant
= (80, 160)


38.
(a)
T1 = 17.5
T2 = 17.5 x (1 - x%)
T3 = 17.5 x (1 - x%)²
......
Tn = 17.5 x (1 - x%)ⁿ⁻¹

Hence, T1, T2, T3,......, T­n is a geometric sequence.
The common ratio = 1 - x%


(b)(i)
The common ratio
= 1 - 28%
= 0.72

Birth rate in 1987, T2
= 17.5 x 0.72
= 12.6 (to 1 decimal place)


Birth rate in 1997, T3
= 17.5 x 0.72²
= 9.07 (to 1 decimal place)


(b)(ii)
Tn : The birth rate in 2017

n = [(2017 - 1977)/10] + 1 = 5

The birth rate in 2017, T5
= 17.5 x 0.72⁴
= 4.7 (to 1 decimal place)