1. Linear sequences
A linear sequence is a list of numbers that have a common difference between each number in the list.
Finding the rule that can extend the sequence from the previous term is called the term to term rule.
This is a basic requirement of sequences but we need to extend our knowledge.

2. Write down the rule and fill in the spaces.
…… ……
…… ……
…… ……

3. Position to term rules
It is important to be able to find the number in a sequence from its position in that sequence.
Example.
If we want to find the 100th term in the sequence
2, 6, 10, 14,18………
Do we need to keep adding 4, until we find the 100th number in the sequence!!
Any suggestions?

4. ………………. Common difference = 4
1, 2, , 4, ……………….Position in sequence (n)
………………. Common difference = 4
4, 8, 12, 16, 20………………. Numbers in 4 x table (4n)
To find numbers in sequence multiply position by 4 and take 2
nth Term = 4n – 2
100th term = (4 x 100) – 2
= 398
2, 6, 10, 14, 18

5. ………………. Common difference = 5
1, 2, 3, 4, ……………….Position in sequence (n)
………………. Common difference = 5
5, 10, 15, 20, 25………………. Numbers in 5 x table (5n)
To find numbers in sequence multiply position by 5 and add 3
nth Term = 5n + 3
100th term = (5 x 100) + 3
= 503
8, 13, 18, 23, 28

6. For each of the number sequences below, find a rule for the nth term (tn) and work out the value of t100.
Question 1
8, 13, 18, 23, 28,
tn= 5n + 3
t100= 5 x = 503
Question 2
1, 4, 7, 10, 13,
tn= 3n - 2
t100= 3 x = 298
Question 3
2, 9, 16, 23, 30,
tn= 7n - 5
t100= 7 x = 695
9, 15, 21, 27, 33,
tn= 6n + 3
t100= 6 x = 603
Question 4
Question 5
-1, 4, 9, 14, 19,
tn= 5n - 6
t100= 5 x = 494
-3, 1, 5, 9, 13,
Question 6
tn= 4n - 7
t100= 4 x = 393
Question 7
6, 18, 30, 42, 54,
tn= 12n - 6
t100= 12 x = 1194

7. Using an nth term to generate a sequence 04 June 2019
If an nth term is given, it is possible to work out any value in the sequence.
It is likely that a question may ask you to generate the first five terms of a sequence from an nth term.
Example
Find the first five terms of the sequence from the following nth terms.
4n + 2
6, 10, 14, 18, 22
7n – 4
3, 10, 17, 24, 31
12n – 5
7,19, 31, 43, 55