Number Patterns

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 …

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.

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.

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.

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.

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 = 5 + 3 5 = 3 + 2 3 = 2 + 1 2 = 1 + 1 1 = 1 + 0

Basic Concept of Functions

… 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

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

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.

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

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.

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

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

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.

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