HKDSE Mathematics Question Review

1. Prove the following arithmetic sum:

2. Show that a recurring decimal is always a rational number.

Solution: Put the sum into a Cartesian plane, we have 11 points (x_n, y_n), where x is number of piles and y is total increment. Now we have (0,0), (1,1), ... (n,n), then the sum is simply the area of a right angle triangle:

Area = xy/2 = （0 + n）n /2

QED

Likewise, if r=1, then Area = (n+1) n /2, which is exactly the sum of arithmetic progression.

Solution:

可見，如果數學卷唔考計算題只考論述題，

相信好多操卷仔會即時跪低，整體合格率大跌

Erratum:

Get stuck with the following value?

Multiply both the numerator and denominator by 10^k, and you are done.

幅圖打錯，正確應為：

Now we visualize the sum:

Take n=5, we have the following blocks:

y

□ □ □ □ ■

□ □ □ ■ ■

□ □ ■ ■ ■

□ ■ ■ ■ ■ x

Then x-y =1

The sum is half of the rectangle,

I.e. sum = xy/2 = y (y+1) /2