Unit 10 Section 3 : Second Differences and Quadratic Sequences

Example 1

(a)

Calculate the first 6 terms of the sequence defined by the quadratic formula,

u_n = n^2 + n – 1

Substituting n = 1 gives,

u₁	= 1² + 1 – 1
	= 1

For n = 2,	u₂	= 2² + 2 – 1
		= 5

For n = 3,	u₃	= 3² + 3 – 1
		= 11

For n = 4,	u₄	= 4² + 4 – 1
		= 19

For n = 5,	u₅	= 5² + 5 – 1
		= 29

For n = 6,	u₆	= 6² + 6 – 1
		= 41

So the first 6 terms are,

1, 5, 11, 19, 29, 41

(b)

Calculate the first differences between the terms.

The differences can now be calculated,

...

(c)

Comment on the results you obtain.

Note that the differences between the first differences are constant. They are all equal to 2. These are called the second differences, as shown below.

Sequence	1	5	11	19	29	41	...

First differences	4	6	8	10	12

Second differences	2	2	2	2

Example 2

Calculate the first 5 terms of the sequence defined by the quadratic formula

(a)

Calculate the first 6 terms of the sequence defined by the quadratic formula,

u_n = 3n^2 – n – 2

For n = 1,	u₁	= 3 × 1² – 1 – 2
		= 3 – 1 – 2
		= 0

For n = 2,	u₂	= 3 × 2² – 2 – 2
		= 8

For n = 3,	u₃	= 3 × 3² – 3 – 2
		= 22

For n = 4,	u₄	= 3 × 4² – 4 – 2
		= 42

For n = 5,	u₅	= 3 × 5² – 5 – 2
		= 68

So the sequence is,

2, 8, 22, 42, 68, ...

(b)

Determine the first and second differences for this sequence.

The differences can now be calculated,

Sequence	0	8	22	42	68	...

First differences	8	14	20	26

Second differences	6	6	6

(c)

Comment on your results.

Again, the second differences are constant; this time they are all 6.

Note: For a sequence defined by a quadratic formula, the second differences will be constant and equal to twice the number of n² .

For example,

u_n = n^2 + n – 1 Second difference = 2
u_n = 3n^2 – n – 2 Second difference = 6
u_n = 5n^2 – n + 7 Second difference = 10

Example 3

Determine a formula for the general term of the sequence,

2, 9, 20, 35, 54, ...

Consider the first and second differences of the sequence:

...

As the second differences are constant and equal to 4, the formula will begin

u_n = 2n^2 + ...

To determine the rest of the formula, subtract 2n^2 from each term of the sequence, as shown below:

Sequence	2	9	20	35	54	...
2n^2	2	8	18	32	50

New sequence	0	1	2	3	4

The new sequence has a constant difference of 1 and begins with 0, so for this sequence the formula is n – 1.

Combining this with the 2n^2 gives

u_n = 2n^2 + n – 1

Example 4

(a)

Calculate the first and second differences for the sequence,

4, 1, 0, 1, 4, 9, ...

...

-3

-1

(b)

Use the differences to determine the next 2 terms of the sequence.

Extending the sequences above gives,

...

-3

-1

(c)

Determine a formula for the general term of the sequence.

As the second differences are constant and all equal to 2, the formula will contain an 'n²' term, and be of the form

u_n = n^2 + an + b

We must now determine the values of a and b. The easiest way to do this is to subtract n² from each term of the sequence, to form a new, simpler sequence.

our sequence	4	1	0	1	4	9
sequence n²	1	4	9	16	25	36
new sequence	3	–3	–9	–15	–21	–27

The new sequence

–3

–9

–15

–21

–27

...

–6

has constant first differences of –6 so will be given by –6n + b.

Using the first term gives,

3	= –6 × 1 + b
b	= 9

Thus the formula for the simpler sequence is –6n + 9.
Now combining this with the n² term gives,

u_n = n^2 – 6n + 9

Exercises

Note: To type indeces on this page use ^ sign. e.g.n^2 :

Question 4

Determine the formula for the general term of each of the following sequences:

(a)	1, 7, 17, 31, 49, 71, ...	u_n =
(b)	6, 18, 38, 66, 102, 146, ...	u_n =
(c)	–5, 10, 35, 70, 115, 170, ...	u_n =
(d)	1, 10, 25, 46, 73, 106, ...	u_n =

Question 7

Determine the formula for the general term of each of the following sequences:

(a)	3, 17, 39, 69, 107, ...	u_n =
(b)	5, 18, 37, 62, 93, ...	u_n =
(c)	9, 23, 45, 75, 113, ...	u_n =
(d)	–4, 12, 38, 74, 120, ...	u_n =