S1T1 Number Sequences

Number Patterns

Usually in the form of structured series of numbers in uniform changes

Recognising Number Sequences

Consider the number sequence 4, 7, 10, 13, 16, 19, . . .

The first term in the sequence is 4, and the second term is 7.

How can we know the 'pattern' to the next number after 19 ?

Usually, we look at the first three numbers 4, 7 and 10, and check for a pattern. 

The most common way is to check for the  pattern is to find the difference between them. 

        = > 4 to 7 = 3, 7 to 10 = 3

=> the next sequence is adding 3.

Therefore the number after 19 is 19 + 3 = 22.

Example

Find the next three terms of the number sequence 3, 7, 11, 15, …

        7 -3 = 4, 11 - 7 = 4 (Step 1 : Find sequence pattern with first 3 numbers)

+4 is the pattern (Step 2 : Add to sequence 'pattern'

15 + 4 = 19, 19 + 4 = 23, 23 + 4 = 27

The next three terms are 19, 23 and 27.

Common Words used

Position(Term)           1     2      3     4      5 .... N  
Pattern                      2 ,   5  ,   8  , 11 , 14 .... nth term 
                                     +3    +3   +3    +3       

a = The first term = The first number = a = 2

d = difference between 2 numbers = 3 

Nth term = the value at position N

Formula/Equation:   Nth term = a + (n - 1)d

Example: What is the value of the 11th term ?

  15th term = 2 + (11 - 1) x 3 = 32

Evaluating the "n^th" Term (the "position" in the sequence) 

A formula or Expression can be given for a number sequence.

Then we can find the number in the sequence by substituting or "putting" the value into the Expression.

Example

A number sequence has the expression 2n + 3. Find the first three terms.

First term => n = 1

  5n + 2

When n = 1,     (Step 1 : Write Expression, and substitute )

    5n + 2  =5(1) + 2             (Step 2 : Substitute value to find answer)

    = 7

When n = 2,

    5n + 2 = 5(2) + 3 =13

When n = 3, 

               5n + 2 = 5(3) + 3 = 18

The first 3 terms are 7, 13, 18

Example

The n^th term of a number sequence is 4n - 2. Find the 6^th and 7^th term.

4n - 2 (Step 1 : Write Expression, and substitute )

When n = 6, 

4n - 2 = 4(6) - 2 = 22 (Step 2 : Substitute value to find answer)

When n = 7, 

4n - 2 = 4(7) - 2 = 28 - 2 = 26

The 6^th term is 22 and the 7^th term is 26

Forming the Number Sequence Expression 

Example

The first 4 terms of a sequence are 4, 11, 18, 25

 a.  Find an expression for the nth term of this sequence

                 4 n + 7

     Step 1 : Find the pattern 

11 - 4 = 18 - 11 = 7 => pattern = +7

     

     Step 2: Form the first term with ? + 7n = 4

  1st term => n = 1

? + 7(1) = 4

     Step 3: Write the Expression

nth term = -3 + 7n

Practice

1.  Find the next three terms of the following number sequence

a.  1, 5, 9, 13, …

b.  7, 10, 13, 16,...

c.  102, 107, 112, 117, …

2.   Nt^h term of a number sequence is 3n - 1

(a)   Find the first three terms

(b)   Find the 10^th term

(c)   Find the 99^th term

3.  Simplify 2x  - 3(x - 2) [12/I/17/2/A]

                   3           5

4.  Simplify 3x  - 2x + 1 [14/II/17/8/A]

                    2         3

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