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标题: gwd 26-29 [打印本页]

作者: pimplekid 时间: 2006-5-27 08:22
标题: gwd 26-29

Q29:

If the sequence x₁, x_2,x_{3, …,}x_n, … is such that x₁ = 3 and x_n₊₁
= 2x_n – 1 for n ≥ 1, then x₂₀– x₁₉ =

A. 2¹⁹

B. 2²⁰

C. 2²¹

D. 2²⁰- 1

E. 2²¹- 1

我怎么都算不出答案，谢谢各位大侠

作者: yaoyao99 时间: 2006-5-27 11:31
X20 - X19
= (2 * X19 - 1) - X19
= X19 - 1
= (2 * X18 - 1) - 1
= 2 * (2 * X17 - 1) - 2
= 2 * (2 * (2 * X16 - 1) - 1) - 2
= 2 * 2 * 2 * X16 - (2 * 2 - 2 - 1) - 1
= 2^18 * X1 - (2^17 - 2^16 - ... - 2^2 - 2^1 - 2^0) - 1
= 2^18 * 3 - sum of(2^k) where k goes from 0 to 17 (see note) - 1
= 2^18 * 3 - 2^0 * (2^18 - 1) / (2 - 1) - 1
= 2^18 * 3 - 1 * 2^18 + 1 - 1
= 2^18 * 2
= 2^19

Note: sum of (x^k) where k goes from y to z = x^y * (x^(z - y + 1) - 1) / (x - 1)

A

作者: pimplekid 时间: 2006-5-31 19:35
谢了，cd真是名不虚传啊，我们这俩人都没作出来。

作者: yaoyao99 时间: 2006-6-1 05:15
It's the formula for geometric series. You just have to know it.

http://mathworld.wolfram.com/GeometricSeries.html

Good luck studying, =)

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