1. Sequences – Linear & Quadratic – Demonstration
This resource provides animated demonstrations of the mathematical method.
Check animations and delete slides not needed for your class.

2. 5, 8, 11, 14, … 3n + 2 Linear Sequences difference +2 +2 +2 nth term
(Arithmetic Sequences)
difference +2 +2 +2
nth term
formula
5, 8, 11, 14, …
3n + 2
1st term
n = 1
4th term
n = 4
2nd term
n = 2
3rd term
n = 3
In a linear sequence, the numbers increase/decrease
by the same amount every time, just like a times table.
We want to find a formula for the nth term.
n = the position of the number (5th, 6th, 20th, 1000th)

3. 5, 7, 9, 11, … 2 4 6 8 +2 +2 +2 +3 nth term formula = (2 × n) + 3
Example 1 +2 +2 +2
5, 7, 9, 11, … +3 2 4 6 8
1) What times table is hidden in the sequence?
2) What do we need to add/subtract
to make the sequences match?
nth term formula =
(2 × n)
+ 3
= 2n + 3
CHECK!
n = 4
4th = (2 × 4) + 3 = 11

4. 6, 10, 14, 18, … 4 8 12 16 +4 +4 +4 +2 nth term formula = (4 × n) + 2
Example 2 +4 +4 +4
6, 10, 14, 18, … +2 4 8 12 16
1) What times table is hidden in the sequence?
2) What do we need to add/subtract
to make the sequences match?
nth term formula =
(4 × n)
+ 2
= 4n + 2
CHECK!
n = 3
3rd = (4 × 3) + 2 = 14

5. Example 2
Your Turn +4 +4 +4 +3 +3 +3
6, 10, 14, 18, …
5, 8, 11, 14, … +2 +2 4 8 12 16 3 6 9 12
1) What times table is hidden in the sequence?
2) What do we need to add/subtract
to make the sequences match?
nth term formula =
(4 × n)
+ 2
nth term formula =
(3 × n)
+ 2
= 4n + 2
= 3n + 2
CHECK!
n = 3
CHECK!
n = 2
3rd = (4 × 3) + 2 = 14
2nd = (3 × 2) + 2 = 8

6. 2, 0, −2, −4, … −2 −4 −6 −8 −2 −2 −2 +4 nth term formula = (−2 × n)
Example 1 −2 −2 −2
2, 0, −2, −4, … +4 −2 −4 −6 −8
1) What times table is hidden in the sequence?
2) What do we need to add/subtract
to make the sequences match?
nth term formula =
(−2 × n)
+ 4
= −2n + 4
= 4 − 2n
CHECK!
n = 4
4th = 4 + (−2 × 4) = −4

7. 2, −1, −4, −7, … −3 −6 −9 −12 −3 −3 −3 +5 nth term formula = (−3 × n)
Example 2 −3 −3 −3
2, −1, −4, −7, … +5 −3 −6 −9
−12
1) What times table is hidden in the sequence?
2) What do we need to add/subtract
to make the sequences match?
nth term formula =
(−3 × n)
+ 5
= −3n + 5
= 5 − 3n
CHECK!
n = 4
4th = 5 + (−3 × 4) = −7

8. Example 2
Your Turn −3 −3 −3 −2 −2 −2
2, −1, −4, −7, …
5, 3, 1, −1, … +5 +7 −3 −6 −9
−12 −2 −4 −6 −8
1) What times table is hidden in the sequence?
2) What do we need to add/subtract
to make the sequences match?
nth term formula =
(−3 × n)
+ 5
nth term formula =
(−2 × n)
+ 7
= −3n + 5
= −2n + 7
= 5 − 3n
= 7 − 2n
CHECK!
n = 4
CHECK!
n = 4
4th = 5 + (−3 × 4) = −7
4th = 7 + (−2 × 4) = −1

9. 1) Find the 2nd difference & halve it to find the n2 coefficient
2) Subtract the quadratic from the original sequence
3) Express the remainder as a linear sequence
4) Join the quadratic with the linear sequence
Quadratic Sequences
+ 2
+ 2
+ 2
Quadratic = 1n2
+ 4
+ 6
+ 8
+ 10
96, 10, 16, 24, 34 n 1 2 3 4 5
Original 6 10 16 24 34
Quadratic: 1n2 9 25
Remainder 7 8
n + 4
1n2 + n + 4

10. 1) Find the 2nd difference & halve it to find the n2 coefficient
2) Subtract the quadratic from the original sequence
3) Express the remainder as a linear sequence
4) Join the quadratic with the linear sequence
Quadratic Sequences
+ 2
+ 2
+ 2
Quadratic = 1n2
+ 5
+ 7
+ 9
+ 11
96, 11, 18, 27, 38 n 1 2 3 4 5
Original 6 11 18 27 38
Quadratic: 1n2 9 16 25
Remainder 7 13
2n + 3
1n2 + 2n + 3

11. 1) Find the 2nd difference & halve it to find the n2 coefficient
2) Subtract the quadratic from the original sequence
3) Express the remainder as a linear sequence
4) Join the quadratic with the linear sequence
Quadratic Sequences
+ 2
+ 2
+ 2
Quadratic = 1n2
+ 1
+ 3
+ 5
+ 7
93, 4, 7, 12, 19 n 1 2 3 4 5
Original 7 12 19
Quadratic: 1n2 9 16 25
Remainder −2 −4 −6
−2n + 4
1n2 − 2n + 4

12. 1) Find the 2nd difference & halve it to find the n2 coefficient
2) Subtract the quadratic from the original sequence
3) Express the remainder as a linear sequence
4) Join the quadratic with the linear sequence
Quadratic Sequences
+ 4
+ 4
+ 4
Quadratic = 2n2
+ 7
+ 11
+ 15
+ 19
96, 13, 24, 39, 58 n 1 2 3 4 5
Original 6 13 24 39 58
Quadratic: 2n2 8 18 32 50
Remainder 7
n + 3
2n2 + n + 3

13. Start by finding 1st / 2nd differences
Calculate whether each sequence is linear, quadratic or neither.
Use the nth term formula to find the value of the 7th term.
6, 10, 14, 18, 22
5, 5, 10, 15, 25
1, 4, 7, 10, 13
6, 10, 16, 24, 34
2, 6, 18, 54, 162
3, 1, −1, −3, −5
3, 3, 5, 9, 15

14. Start by finding 1st / 2nd differences
Calculate whether each sequence is linear, quadratic or neither.
Use the nth term formula to find the value of the 7th term.
6, 10, 14, 18, 22
Linear 4n th term = 30
5, 5, 10, 15, 25
Neither (Fibonacci-type)
1, 4, 7, 10, 13
Linear 3n − th term = 19
6, 10, 16, 24, 34
2, 6, 18, 54, 162
3, 1, −1, −3, −5
3, 3, 5, 9, 15

15. Start by finding 1st / 2nd differences
Calculate whether each sequence is linear, quadratic or neither.
Use the nth term formula to find the value of the 7th term.
6, 10, 14, 18, 22
Linear 4n th term = 30
5, 5, 10, 15, 25
Neither (Fibonacci-type)
1, 4, 7, 10, 13
Linear 3n − th term = 19
6, 10, 16, 24, 34
Quadratic n2 + n th term = 60
2, 6, 18, 54, 162
Neither (Geometric)
3, 1, −1, −3, −5
Linear − 2n 7th term = −9
3, 3, 5, 9, 15
Quadratic n2 − 3n th term = 33

16. tom@goteachmaths.co.uk Questions? Comments? Suggestions?
…or have you found a mistake!?
Any feedback would be appreciated .
Please feel free to