Arithmetic and Geometric

Tieria

2009-11-15 7:43 am

1. The sum of three numbers is 9 and their product is -216. Find the three numbers if they form a geometric sequence.

The answer is 3, -6, 12.

2. At noon on 1st January, a certain biology experiment started with 300 bacteria. If the number of bacteria increased to three times that of the original in every 3 hours, and there were N bacteria at noon on 2nd January,
(a) find the value of N,
(b) when was it that there were N / 3 bacteria?

The answers are (a) 1968300 and (b) at 9 a.m. on 2nd January.

回答 (1)

用戶9112

2009-11-15 7:54 am

✔ 最佳答案

(1) Let the numbers be a, ar and ar^2
a + ar + ar^2 = 9 ... (1)
(a)(ar)(ar^2) = -216 ... (2)
(2) => a^3r^3 = -216
ar = -6 ... (3)
Sub into (1), a - 6 - 6r = 9
a = 15 + 6r ... (4)
Sub into (3), (15 + 6r)r = -6
6r^2 + 15r + 6 = 0
2r^2 + 5 + 2 = 0
(2r + 1)(r + 2) = 0
r = -1/2 or r = -2
Use (4), when r = -1/2, a = 12; the 3 numbers are 12, -6 and 3
when r = -2, a = 3; the 3 numbers are 3, -6, 12 same as above
2. (a) From noon on 1st January to noon on 2nd January, there are 24 hours and thus 8 3-hours period
N = 300 * 3^8
N = 1968300
(b) There were N / 3 bacteria 3 hours before noon of 2nd January. The time is at 9 a.m. on 2nd January

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