Maths. 等差等比數列問題

[匿名]

2015-01-02 4:37 am

Alex draws some straight lines on a piece of paper. At the beginning, he draws a straight line of length 3 cm from a point P(o) to another point P(1). Then, starting from P(1), he draws the second straight line of length 6cm to a point P(2), and so on. P(o),P(1),P(2),.... may not be collinear.

Suppose the length of the n th line drawn is 3(2)^(n-1) cm.

a) Express the total length of the first x straight lines drawn in terms of x.

b) Alex claims that if (x+1) straight lines are drawn, it is possible that the points
P(x+1) and P(0) overlap with each other. Is he correct? Explain your answer.

我唔知到 (b) part 應該點做好...諗左好耐, 請各位幫幫忙，

感激不盡！

回答 (1)

用戶2053

2015-01-02 7:59 am

✔ 最佳答案

a)The total length of the first x straight lines drawn
= 3 + 3(2) + 3(2²) + ... + 3(2ⁿ⁻¹)
= 3(1 - 2ⁿ) / (1 - 2)
= 3(2ⁿ - 1)
b)He is NOT correct.
Because the maximum distance between P(o) and P(x)
= P(o) P(1) + P(1) P(2) + P(2) P(3) + ... + P(x-1) P(x)
(i.e. P(o),P(1),P(2),....,P(x) are collinear)
= 3(2ⁿ - 1) < P(x) P(x+1) = 3(2ⁿ)
Therefore P(x+1) and P(0) overlap with each other is impossible.

2015-01-02 00:26:06 補充：
打錯 x 為 n , 更正:

a)

The total length of the first x straight lines drawn
= 3 + 3(2) + 3(2²) + ... + 3(2ˣ⁻¹)
= 3(1 - 2ˣ) / (1 - 2)
= 3(2ˣ - 1)

2015-01-02 00:26:41 補充：
b)

He is NOT correct.
Because the maximum distance between P(o) and P(x)
= P(o) P(1) + P(1) P(2) + P(2) P(3) + ... + P(x-1) P(x)
(i.e. P(o),P(1),P(2),....,P(x) are collinear)
= 3(2ˣ - 1) < P(x) P(x+1) = 3(2ˣ)
Therefore P(x+1) and P(0) overlap with each other is impossible.

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