### Quadratic Sequences   More

Year 10/11 maths: a quadratic sequence is a set of numbers where the difference between the terms changes.

###### What is a quadratic sequence

A sequence is a string of numbers, that follow a particular pattern. The individual numbers in a sequence are called terms.

In a quadratic sequence, the difference between terms increases, or decreases, at a constant rate.

If the second differences are constant, then it is a quadratic sequence.

Is 1, 2, 4, 7, 10 a quadratic sequence?

Look at the difference between the terms.

The first difference changes as the terms increase.

The second differences are a constant
So it is a quadratic sequence.

Work out whether the following are quadratic sequences:
1. 7, 12, 19, 28, 39
2. 1, 5, 14, 27, 38
3. –10, 4, 12, 38, 74, 120

ANS

###### nth term of quadratic sequence Menu

The nth term of a quadratic sequence is given by:

an2 + bn + c
where a, b and c are numbers and a ≠ 0..

e.g. 2n2 + 3n + 4 where a=2, b=3 and c=4

1st term is when n = 1 which is 2 + 3 + 4 = 9

2nd term (n = 2) is 2 × 2 + 3 × 2 + 4 = 18

3rd term (n = 3) is 2 × 32 + 3 × 3 + 4 = 31

Q. The nth term of a quadratic sequence is 3n2 – 3n + 2

Write down the first three terms

ANS

###### Find 'a' of nth term expression Menu

Look at the quadratic sequence, 7, 11, 19, 31, 47

The first difference between the numbers is 4, 8, 12

The second difference is a constant 4

This lets us work out 'a' of the nth term expression for this quadratic sequence.

If 2nd difference = 2, start with n2   (a = 1)

If 2nd difference = 4, start with 2n2 (a = 2)

If 2nd difference = 6, start with 3n2 (a = 3)

So for 7, 11, 19, 31, 47 the nth term expression starts 2n2

Q. What is 'a' for this quadratic sequence: 1, 4, 13, 28, 49 ?

ANS

###### 'b', 'c' of nth term expression Menu

The nth term of a quadratic sequence is given by:

an2 + bn + c where a, b and c are numbers.

Let's work out 'b' and 'c' for the nth term expression for 7, 11, 19, 31, 47

We already know it starts with 2n2

We know that when n = 1 , the term is 7 and when n = 2, the term is 11.
So we can write two equations:

For n = 1,     7 = 2 × 12 + b + c   → 5 = b + c
For n = 2,  11 = 2 × 22 + 2b + c → 3 = 2b + c

So b = –2 and c = 7; nth term = 2n2 –2n + 7

Find an expression for the nth term of this sequence for 1, 4, 13, 28, 49

ANS