Quod Erat Demonstrandum

2016/08/29

講兩題

Filed under: Junior Form Mathematics,NSS — johnmayhk @ 5:02 下午
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seize-the-moment-geek-green

對,seize the moment,抓緊此際講兩題。

(注:上圖式子是 k^{th} moment 的定義。特別地,當 k=2,它就是方差 variance。)

(一)

修物理的同學一定學過透鏡公式(lens formula)

\frac{1}{f}=\frac{1}{u}+\frac{1}{v} (more…)

廣告

2015/01/08

被 8 整除

Filed under: Fun,mathematics — johnmayhk @ 2:08 下午
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1.問題

證明對於任何正整數 n

n^4+2n^3+3n^2+2n

必可被 8 整除。

2.解答

(a)

本來利用 Mathematical Induction (M.I.) 來證明是很輕易的,但現在的 M2 課程已刪除了整除性,相信中四五的同學,要解題不易。

只寫 P(k)\Rightarrow P(k+1) 這步: (more…)

2012/12/31

費波那契數之和

Filed under: Pure Mathematics — johnmayhk @ 4:33 下午
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以下是費波那契數列(Fibonacci sequence):

F_1=1
F_2=1
F_3=F_2+F_1=2
F_4=F_3+F_2=3
F_5=F_4+F_3=5
F_6=F_5+F_4=8

… …

原來「任何正整數皆可寫成若干不同的(distinct)費波那契數之和」, (more…)

2009/05/13

數數唸

一.

以下數字是「旋轉對稱」的嗎?

1961 (more…)

2009/01/02

Something about F.6 Pure Math First Term Exam

Filed under: HKALE,Pure Mathematics — johnmayhk @ 4:24 下午
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Here is one of questions:

Given that

x_1 = 4, x_2 = 12
x_{n + 2} = 4(x_{n + 1} - x_{n}) (\forall n \in \mathbb{N})

Prove that

(a) x_n = 2(1 + \frac{1}{n})x_{n - 1}
(b) x_n = (n + 1)2^n

The question requires students to use mathematical induction to prove that.

I’d like to give another ways.

(method 1) (more…)

2008/11/21

Exist or does not exist

Is giving hints a good way to help students in solving mathematics problems? Urm, sometimes it may not.

Here is a common m.i. question in recent F.4 additional mathematics regular test:

Show that n^3 - n + 3^n is divisible by 3 for any positive integer n. (more…)

2008/11/18

Create an m.i. question

It is not difficult to create questions like:

Prove by mathematical induction that

\frac{3^3\times1}{4!} + \frac{3^4\times2}{5!} + \frac{3^5\times3}{6!} + \dots + \frac{3^{n+2}\times n}{(n+3)!} = \frac{9}{2} - \frac{3^{n+3}}{(n+3)!} (more…)

2008/10/15

Assuming step in mathematical induction

Just share a minor point in the presentation of M.I.

To prove that a proposition P(n) is true for all positive integers n by using M.I.

We need ‘4’ steps, namely (more…)

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