Here are the nth terms for the given sequences: (a) The nth term is: 3n + 1 (b) The nth term is: n + 2 (c) The nth term is: n + 1 (d) The nth term is: 10n (e) The nth term is: 5n - 1

1. Number sequencesBTEOTSSSBAT write down the next term in a sequence and describe sequences using the nth term

2. Key termsTermSequenceDifference methodFormulanth term

3. A number sequence is a set of numbers with a rule to find every number (term) in the sequence.For example: we can describe a sequence by n + 1. INPUTOUTPUTn + 1

4. The letter ncan represent any number – in For example 1, 2, 3 and 4. These are then the first four terms of the n = 1 sequence.INPUTOUTPUT1n + 12

5. The letter ncan represent any number – in For example 1, 2, 3 and 4. These are then the first four terms of the n = 1 sequence.INPUTOUTPUT2n + 13

6. The letter ncan represent any number – in For example 1, 2, 3 and 4. These are then the first four terms of the n = 1 sequence.INPUTOUTPUT3n + 14

7. The letter ncan represent any number – in For example 1, 2, 3 and 4. These are then the first four terms of the n+ 1 sequence.INPUTOUTPUT4n + 15

8. We can put these results in a table:The sequence 2,3,4,5 can be called the n + 1 sequence

9. Write down the sequence of 2nINPUTOUPUT12342n

10. Write down the sequence of 2nINPUTOUPUT123424682n

11. The difference methodOne method to find the next terms and also to find a formula to describe the sequence is the difference method.1 3 5 792222The difference between each number is 2, so the next number will be 7 + 2 = 9; the next one after that will be 9 + 2. The rule for this sequence is add 2.

12. Now try these 1Find the next two numbers in the sequence and a describe the rule for the sequence(a) 5, 10, 15, 20, ...(b) 6, 5, 4, 3, 2.....(b) 4, 8, 12, 16, ...(c) 1, 3, 6, 10, .....(d) 5, 10, 20, 40, .....(e) 50, 40, 30, 20 .....

13. Finding the nth termThe difference method can be used to find a rule to find any term. ExampleFind the nth term for the sequence 4, 8, 12, 16, ...

14. ExampleFind the nth term for the sequence 4, 8, 12, 16, ...SolutionThe difference is 4– this tells us that the nth term with involve 4nWe now need to see what the sequence 4n looks like and compare it with the original sequenceThere is no difference between 4n and the original sequence. So our sequence is 4n. The nth term is 4n.

15. ExampleFind the nth term for this sequence: 5, 8, 11, 14 and hence find the 20th term.The difference is 3; this tells us that the nth term will involve 3n.3n is 2 short of the original sequence. So, the nth term will be 3n + 2.The 20th term means that n = 20:3 20 + 2 = 62.The 20th term is 62.

16. Now try these 2Find the nth term for each of the following number sequences:(a) 1, 5, 9, 13, ...(b) 7, 9, 11, 13 ...(c) 4, 5, 6, 7, ...(d) 11. 21, 31, 41, ...(e) 6, 11, 16, 21, ...