Number Patterns

1. Investigate number patterns to make and test conjectures 2. Generalise relationships in number patterns 3. Investigate and generalise number patterns where there is a constant difference between consecutive terms and the general term is therefore linear

1. Investigate number patterns to make and test conjectures A conjecture is a statement which, although evidence can be found to support it, has not been proved to be true or false. Example: Conjecture: The sum of the first n odd numbers equals the square of n 1=1 1+3 = = =16

Continued The answers represent the square of: 1 odd number = 2 odd numbers The next step will be to find the general proof for this statement or to show that it is incorrect by using a counter proof.

Test Your Knowledge Look at the pattern below: = = = Make a conjecture about the last number in the 5th row. Test your conjecture by writing the next two rows. What will the last number be at both ends of the 10th row?

Solutions = = and 120

2. Generalise relationships in number patterns Number patterns are a regular occurrence in Mathematics. Ordered list of numbers are called a sequence and the numbers we referred to as terms with symbol:, where It is useful to determine a general rule in order to determine the next or any other term of the sequence.

Continued This can be in the form of a recursive pattern (each new term is defined in relation to some terms which have been made previously) e.g. 3; 5; 7; 9; ……….. We see that by adding 2 we can determine the next term and make the conjecture that the general term can be determined using the rule: We refer to the constant adding of 2 as a common difference between terms

3. Investigate and generalise number patterns where there is a constant difference between consecutive terms Look at the following patterns of numbers: 5; 10; 15; ………… We want to know what the next three numbers / terms of this sequence will be as well as the general term of the sequence. Each sequence is a group of numbers that has two very important properties: o the terms are listed in a specific order and o there is a rule which enables you to continue with the sequence. Either the rule is given or you have to determine it using the first three terms of the sequence.

Continued Consider the sequence: 3 ; 5 ; 7 ; 9. Using the general formula Tn = an+b where a is the constant difference between successive terms and b is the value for To Constant difference = 2 - Constant diff = linear number pattern T n = 2n + b 3 = 2(1) + b b =1 Tn = 2n+1

Worked Example Given: 4; 9; 14; 19;……. Note there is a common difference (a) of 5 between successive terms. Tn = 5n + b 4 = 5(1) + b b = -1 Tn = 5n - 1 If there exists a first common difference between successive terms, it is called a linear pattern

Test Your Knowledge Question 1 Determine the general term of the sequence: 1; 5; 9; 13; 17;……. ……… Answer A) Tn = 4n + 2 B) Tn = 5n + 1 C) Tn = 4n-3D) Tn =5n-1

5. Use graphs to represent number patterns If we again look at the example 1; 5; 9; 13; 17; 21; ……. T1 = 1 T2 = 5 T3 = 9 Tn = 4n-3 This represents a linear discrete graph where the difference between consecutive terms are constant. If points are discrete it means they are not joined with a line as

Test Your Knowledge 1. Plot the graph of the following sequence: 21; 18; 15; 12;………..

Solution

Bibliography Examples from Oxford “Mathematics Plus” Grade 10