# Quod Erat Demonstrandum

## 2016/08/29

### 講兩題

Filed under: Junior Form Mathematics,NSS — johnmayhk @ 5:02 下午
Tags: , ,

（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）

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

## 2015/01/08

### 被 8 整除

Filed under: Fun,mathematics — johnmayhk @ 2:08 下午
Tags: , ,

1.問題

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

2.解答

(a)

## 2012/12/31

### 費波那契數之和

Filed under: Pure Mathematics — johnmayhk @ 4:33 下午
Tags:

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

… …

1961 (more…)

## 2009/01/02

### Something about F.6 Pure Math First Term Exam

Filed under: HKALE,Pure Mathematics — johnmayhk @ 4:24 下午
Tags: ,

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…)