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Linear Number Sequences/Patterns
A linear number sequence is a sequence of numbers that has a constant
difference between adjacent terms. Consider the first five terms of the
number sequence shown: The difference between adjacent terms is 3.
position 
1st
terms 
5, 8, 11, 14, 17,………………..?
difference 
+2
2nd
3
3rd
3
4th
3
5th................................nth
3
We want to obtain a general rule that gives us the value of any term (nth) in the
sequence as a function of the term’s position.
n
3n
3n?+ 2
Can you see how the numbers of this sequence
5
1
3
are related to those in the 3 times table?
8
2
6
Adjacent numbers in the 3 times table also differ
by 3. The terms in this sequence are 2 bigger
than the numbers in the 3 times table.
tn= 3n + 2
3
9
11
4
12
14
5
15
17
1st
2nd
3rd
4th
5th.........nth
5, 8, 11, 14, 17,…..?
+2
3
3
3
3
3, 7, 11, 15, 19,…..?
-1
4
4
4
tn= 3n + 2
4
tn= 4n - 1
n
4n
4n?- 1
1
4
3
2
8
7
3
12
11
4
16
15
5
20
19
1st
2nd
3rd
4th
5th.........nth
5, 8, 11, 14, 17,…..?
+2
3
3
3
3
3, 7, 11, 15, 19,…..?
-1
4
4
4
4
8, 13, 18, 23, 28,…..?
+3
5
5
5
tn= 3n + 2
5
tn= 4n - 1
tn= 5n + 3
n
5n
5n?+ 3
1
5
8
2
10
13
3
15
18
4
20
23
5
25
28
1st
2nd
3rd
4th
5th.........nth
5, 8, 11, 14, 17,…..?
+2
3
3
3
3
3, 7, 11, 15, 19,…..?
-1
4
4
4
4
8, 13, 18, 23, 28,…..?
+3
5
5
5
5
-1, 1, 3, 5, 7,…..?
-3
2
2
2
tn= 3n + 2
2
tn= 4n - 1
tn= 5n + 3
tn= 2n - 3
1. The common difference tells you the multiple of n required for the first part
of the rule.
2. The second part of the rule is obtained by subtracting the first term and
the common difference.
2a. This is equivalent to asking yourself what you need to do to the common
difference to get to the value of the first term.
Example Question 1
For the number sequence below:
(a) Find the “position to term” rule (b) Use your rule to find the 58th term (t58)
2, 9, 16, 23, 30,……
(a) tn= 7n - 5
Difference 7  7n
72-5
(b) t58= 7 x 58 - 5 = 401
Example Question 2
For the number sequence below:
(a) Find the “position to term” rule (b) Use your rule to find the 75th term (t75)
9, 15, 21, 27, 33,……
(a) tn= 6n + 3
(b) t75= 6 x 75 + 3 = 453
Difference 6  6n
69+3
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
Question 2
1, 4, 7, 10, 13,
tn= 3n - 2
t100= 3 x 100 - 2 = 298
Question 3
2, 9, 16, 23, 30, tn= 7n - 5
t100= 7 x 100 - 5 = 695
Question 4
9, 15, 21, 27, 33, tn= 6n + 3
t100= 6 x 100 + 3 = 603
Question 5
-1, 4, 9, 14, 19,
tn= 5n - 6
t100= 5 x 100 - 6 = 494
Question 6
-3, 1, 5, 9, 13,
tn= 4n - 7
t100= 4 x 100 - 7 = 393
Question 7
6, 18, 30, 42, 54, tn= 12n - 6
t100= 5 x 100 + 3 = 503
t100= 12 x 100 - 6 = 1194
Can you suggest why they are called linear sequences?
10
8
6
tn= 2n + 1
4
2
tn= 3n - 4
0
1
2
3
4
5
Number sequences can be
used to solve problems
involving patterns in diagrams.
4
3
2
1
1
3
S = 2D - 1
How many
squares of
chocolate (S) will
the 10th diagram
(D) contain?
5
7
S10 = 2 x 10 - 1 = 19
How many wooden braces (B) will there be, in the 20th panel (P)?
1
2
3
4
7
10
B = 3P + 1
B20 = 3 x 20 +1= 61
3
How many stone
slabs (S) will the
15th diagram (D)
contain?
2
1
1
S = 4D - 3
5
9
S15 = 4 x 15 – 3 = 57
How many steel braces (B) will there be, in the 28th panel (P)?
1
6
B = 5P + 1
2
11
B28 = 5 x 28 + 1 = 141
3
16