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SEQUENCE AND PATTERNS
Introduction:
We often need to spot a pattern in order to predict what will happen next. In maths, the correct name for a pattern of numbers is called a SEQUENCE. In this topic therefore you will learn how to identify and describe general rules for patterns. You will be able to determine a term in the sequence and find the missing numbers in the sequence .
Draw and Identify the Patterns
For any pattern it is important to try to spot what is happening before you can predict the next number.
Activity : Identifying the number patterns
(a) 3,6,9,12 . . .
To obtain the next number in the sequence, we add 3 to the previous number. The numbers in this sequence are multiples of 3.
(b) 7,14,21,28 . . .
To obtain the next number in the sequence, we add 7 to the previous number. The numbers in this sequence are multiples of 7.
Describing the General Rule
Activity: Finding the Next Term in the sequence
Find the next numbers in the sequences below
Exercise
Generating Number Sequence
Activity : Generating a sequence
This involves using a formulae to generate sequences for given values.
EXAMPLES
What sequence do you generate by using the following formula? Take n = 1,2,3,4,5 . . .
1. 2n
we substitute the value of n ,in the formula given
2. 8n - 5
we substitute the value of n ,in the formula given
3. 6n+2
we substitute the value of n ,in the formula given
Input/Output machine
Some math problems contain a pattern, so once you nd a pattern then you can make a rule that will solve the problem for a given input. Therefore we put numbers into the machine[input], and the machine uses an operation (add, subtract, multiply or divide) to give us a result[output].
What number comes out of each of these number machines?
Formulae for General Terms
Activity : Identifying the nth term In your groups work in pairs.
Note:
It is very helpful not only to be able to write down the next few terms in a sequence, but also to be able to write down, for example, the 100th or even the 1000th term.
Example:
For the sequence 3, 7, 11, 15, …, …
Find:
a) the next three terms.
b) the 100th term.
c) the 1000th term.
Answer
a) You can see that 4 is added each time to get the next term. So you obtain 19, 23, 27.
b) To find the 100th term, starting at 3, you add 3 to 4 times ninety nine times giving:
3 + 4 x 99 = 3 + 396 = 399
c) Similarly, the 1000th term is
3 + 4 x 999 = 3 + 3996 = 3999
I can go one step further and write down the formula for a general term, i.e. the nth term.
This is 3 + 4 x (n – 1) = 3 + 4n - 4
= 4n – 1.
Exercise.
For each sequence, write down the difference between each term and formula for the nth term.
a) 3, 5, 7, 9, 11, …
b) 5, 11, 17, 23, 29, …
c) 4, 7, 10, 13, 16, …
d) 2, 5, 8, 11, 14, …
e) 6, 10, 14, 18, 22, …
2. a) Write down the first 6 multiples of 11.
b) What is the formula for the nth term of the sequence of the multiples of 11?
c) What is the formula for the nth term of this sequence?
Situation of Integration
There is a family in the neighbourhood of St. Joseph College Ombaci. The family has a rectangular compound on which they want to put up a hedge around. The hedge shall be made up of plants of different colours.
Support: Physical instruments like hoes, machetes, tape measure
Resources: Knowledge of construction of figures like rectangles, patterns, sequences
Task: The family requests you to plant the hedge around their rectangular compound so that it looks beautiful.
Explain how you will plant the hedge, making sure that the plants at the corners of the compound are the same in terms of colour. Discuss whether there are other ways of planting the hedge.
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