1. Number Patterns

2. I played a game and the points I got in each round is shown below…1 2 3 4 5
Points 9 10 5 6 8
The group of numbers 9, 10, 5, 6, 8 forms a sequence.
5th term
1st term
2nd term …

3. forms the sequence of square numbers.
The terms in some sequences may form patterns. Let’s consider some common sequences. … 1 4 9 16 25
The number of dots in each square: 1, 4, 9, 16, 25 are square numbers.
The group of numbers 1, 4, 9, 16, 25, …
forms the sequence of square numbers.

4. Triangular Numbers … 15 1 3 6 10
The number of dots in each triangle: 1, 3, 6, 10, 15 are triangular numbers.
The group of numbers 1, 3, 6, 10, 15, … forms the sequence of triangular numbers.

5. Arithmetic Sequences Consider the following sequence.
3, 6, 9, 12, 15, …
In this sequence, the result obtained by subtracting any term from its following term is a constant (i.e. 3).
6 – 3 = 9 – 6 = 12 – 9 = 15 – 12 = 3
3, 6, 9, 12, 15, …
Such a sequence is called an arithmetic sequence.

6. Geometric Sequences Consider the following sequence.
2, 4, 8, 16, 32, …
In this sequence, the result obtained by dividing any term (except the first term) by its preceding term is a constant (i.e. 2).
2, 4, 8, 16, 32, …
Such a sequence is called a geometric sequence.

7. Fibonacci Sequence The Fibonacci sequence is
1, 1, 2, 3, 5, 8, 13, 21, 34, …
In this sequence, starting from the third term, each term is equal to the sum of the two preceding terms.
Each number in the sequence is called a Fibonacci number.
i.e.
8 = = = = = 1 + 0

8. Basic Concept of Functions

9. … Consider the sequence of square numbers. 1 4 9 16 25
No. of dots in each row 1 2 3 4 5
Total no. of dots 1 4 9 16 25

10. 3rd term = 32 4th term = 42 5th term = 52
Note that:
1st term
= 12
2nd term = 22
3rd term = 32 4th term = 42 5th term = 52
and so on.
No. of dots in each row 1 2 3 4 5
Total no. of dots 1 4 9 16 25

11. General term of a Sequence
We can use the algebraic expression n2 to represent the nth square number or the nth term of the sequence,
i.e. 12, 22, 32, 42, 52, 62, …, n2, …
The nth term, i.e. n2, is called the general term of the sequence of square numbers. It can be used to find any term in the sequence.

12. each term is a power of 3: 31, 32, 33, 34, 35, …
Let’s consider the general terms of the following sequences:
4, 8, 12, 16, 20, 24, 28, …
3, 9, 27, 81, 243, …
each term is a multiple of 4: 4(1), 4(2), 4(3), 4(4), 4(5), 4(6), 4(7), …
each term is a power of 3: 31, 32, 33, 34, 35, …
the general term = 3n
the general term = 4n

13. Follow-up question Find the general terms of the following sequences.
(a) 2, 4, 6, 8, 10, 12, … (b) 2, 4, 8, 16, 32, 64, …
(a) Each term in the sequence is a multiple of 2:
2(1), 2(2), 2(3), 2(4), 2(5), 2(6), …
Therefore, the general term of the sequence is 2n.
(b) Each term in the sequence is a power of 2:
21, 22, 23, 24, 25, 26, …
Therefore, the general term of the sequence is 2n.

14. The general term of a sequence is like a number machine.
substitute a number into the general term
corresponding term of the sequence
general term
like a number machine
input
output

15. For a sequence with general term n + 3…
corresponding term of the sequence is 4
substitute n = 1 into the general term 1 4
For a sequence with general term 1 – n2…
1 – n2
corresponding term of the sequence is –8
substitute n = 3 into the general term 3 –8

16. Note that for each value of x, there is a corresponding value of y.
Suppose two variables x and y have the following relationship:
Input (x) –3 –2 –1 1 2 3 …
Output (y) –6 –4 –2 2 4 6 …
Note that for each value of x, there is a corresponding value of y.
We called y a function of x. It is written as y = 2x.

17. Follow-up question
According to the number machine 2x + x2, answer the following questions.
Number machine
2x + x2
Input x
Output y
(a) Write down a function relating x and y.
(b) Hence, find the value of y when x = 5.
(a) y = 2x + x2
(b) When x = 5, y = 2(5) + 52 = 35