MATHEMATICSGenerating Number Sequences

Aims of the Lesson

• To investigate linear number sequences.

• To learn how to generate sequences from…• written rules• number/function machines• practical situations• expressions for the nth term

Written Rules

• You will be given a start value and the rule to follow in order to generate your sequence.

• E.g. Starting with 6, add on 3

6

• E.g. Starting with 8, double then subtract 1

8

9 12 18…15

15 29 57 113…

Practice

Write down the first 5 terms in each sequence…

1) Starting with 15, subtract 2

2) Starting with 20 000, divide by 10

3) Starting with 7, subtract 1 then double

Answers

1) Starting with 15, subtract 2

15,

2) Starting with 20 000, divide by 10

20 000,

3) Starting with 7, subtract 1 then double

7,

13, 11, 9, 7…

2 000, 200, 20, 2…

12, 22, 42, 82…

Number/Function Machines…

• You are given the operations to undertake on your number(s) in a number or function machine.

• For term-to-term machines, you will be given the first number to input (this will also be the first number of your sequence!).

• For position-to-term machines, you will need to enter 1, 2, 3… into the machine and only the outputs will form the number sequence.

Term-to-Term Example

in out

1 × 8 – 5

• 1 in × 8 8 – 5 = 3• 3 in × 8 24 – 5 = 19• 19 in × 8 152 – 5 = 147 and so on…

• Sequence: 1, 3, 19, 147…

Practice

• Find the first 5 terms in the sequences generated by these term-to-term function machines…

1) 5 + 4

2) 3 – 1 × 5

3) 4 + 16 ÷ 2

Answers

1) Start with 5, add 4

5, 9, 13, 17, 21…

2) Start with 3, subtract 1 then multiply by 5

3, 10, 45, 220, 1095…

3) Start with 4, add 16 then divide by 2

4, 10, 13, 14.5, 15.25…

Position-to-Term Example

in out

× 2 + 3

• For the 1st number, put 1 in × 2 then + 3 5• For the 2nd number, put 2 in × 2 then + 3 7• For the 3rd number, put 3 in × 2 then + 3 9 and so on…

• Sequence (only outputs): 5, 7, 9…

Practice

• Find the first 5 terms in the sequences generated by these position-to-term function machines…

1) + 4

2) – 1 × 5

3) + 16 ÷ 2

Answers

1) add 4:

1+45, 2+46, 3+47, 4+48, 5+49

5, 6, 7, 8, 9…

2) subtract 1 then multiply by 5:

(1-1)x50, (2-1)x55, (3-1)x510, (4-1)x515…

0, 5, 10, 15, 20…

3) add 16 then divide by 2:

(1+16)/28.5, (2+16)/29, (3+16)/29.5…

8.5, 9, 9.5, 10, 10.5…

Sequences from Practical Situations

• A context will be described – very often this involves generating strings of shapes using matches! – and an example given.

• You will need to carry on with drawing the next few shapes in the sequence.

• You will be required to turn the shapes into numbers (usually number of shapes / number of matches).

Example• Here are the first few shapes in a sequence made using

matchsticks. Draw the next two shapes and complete the table of results.

No. of Squares 1 3 5

No. of Matches 4 10 16

Example• Each time, a square is being added to the left AND to the

bottom, so lets continue…

No. of Squares 1 3 5

No. of Matches 4 10 16 7

22

9

28

Practice• Copy the first three shapes.• Draw the next two shapes in the sequence.• Copy and complete the table for the sequence.

No. of Squares

No. of Matches

Answers• Copy the first three shapes.• Draw the next two shapes in the sequence.• Copy and complete the table for the sequence.

No. of Squares

No. of Matches 2

7

4

12

6

17

8

22

10

27

Sequences from Nth term Expression

• You may be given an algebraic expression for the nth term and asked to generate the first few terms of the sequence.

• ‘n’ represents the position of the term and the expression tells you how to work out the value of that term using its position.

• For the 1st number, substitute ‘n’ with the number 1 and evaluate the resultant calculation.

• For the 2nd number, substitute ‘n’ with the number 2 and evaluate the resultant calculation and so on…

Nth Term Example

• Find the first 4 terms of the sequence generated by the expression:

• For the 1st number, put 1 in 4 × 1 + 1 5• For the 2nd number, put 2 in 4 × 2 + 1 9• For the 3rd number, put 3 in 4 × 3 + 1 13 • For the 4th number, put 4 in 4 × 4 + 1 17

• Sequence: 5, 9, 13, 17…

4n + 1

Practice

• Find the first 5 terms of the sequences generated by these expressions…

1) 2n + 3

2) 5n – 2

3) 10 – n

Answers

1) 2n + 3

Ans: 5, 7, 9, 11, 13

2) 5n – 2

Ans: 3, 8, 13, 18, 23

3) 10 – n

Ans: 9, 8, 7, 6, 5

2n + 32 × 1 + 3 = 52 × 2 + 3 = 72 × 3 + 3 = 92 × 4 + 3 = 112 × 5 + 3 = 13 5n – 25 × 1 – 2 = 35 × 2 – 2 = 85 × 3 – 2 = 135 × 4 – 2 = 185 × 5 – 2 = 2310 – n 10 – 1 = 910 – 2 = 810 – 3 = 710 – 4 = 610 – 5 = 5

What next?

Make appropriate notes (including examples) on generating sequences or printout the prepared notes and complete all the tasks within them.

Work through the MyMaths lesson (and then its online homework) called: Algebra > Sequences > Sequences

Now move on to the Seq-Terms powerpoint.