Unit 10 Section 3 : Second Differences and Quadratic Sequences

In section 10.2 we dealt with sequences where the differences between the terms was a constant value. In this section we extend this idea to sequences where the differences are not constant.

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,

1511192941...
4681012
(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.

Sequence1511192941...
First differences4681012
Second differences2222

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,

Sequence08224268...
First differences8142026
Second differences666
(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:

29203554...
7111519
444

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 29203554...
2n^2 28183250
New sequence01234
1111

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, ...

410149...
-3-1135
2222
(b)

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

Extending the sequences above gives,

4101491625...
-3-113579
222222
(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 sequence410149
sequence n2149162536
new sequence3–3–9–15–21–27

The new sequence

3–3–9–15–21–27...
–6–6–6–6–6

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 1

(a)

Calculate the first 6 terms of the sequence defined by,

u_n = n^2 + 2n + 1

, , , , ,
(b)

Calculate the second differences for the sequence.

The second differences:
(c)

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

,
Question 2

A sequence has its general term defined as,

u_n = 8n^2 – n – 1

(a)

What would you expect to be the second differences for the sequence?

(b)

Calculate the first 5 terms of the sequence.

, , , ,
(c)

Calculate the second differences for the sequence.

Question 3

A sequence is listed below:

6, 9, 14, 21, 30, 41, ...

(a)

Calculate the second differences for the sequence.

(b)

Determine the formula for the general term of the sequence.

Remember: To type indeces use ^ sign. e.g.n^2 :

u_n =
Question 4

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

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

A sequence is listed below:

2, 9, 20, 35, 54, 77, ...

(a)

Calculate the second differences for this sequence.

(b)

Form a simpler sequence by subtracting 2n2 from each term.

, , , , ,
(c)

Determine a formula for the general term of the simpler sequence.

Simpler sequence =
(d)

Determine a formula for the general term of the original sequence.

u_n =
Question 6

(a)

Calculate the second differences of the sequence,

6, 17, 36, 63, 98, 141, ...

(b)

Determine the formula for the general term of the sequence.

u_n =
Question 7

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

(a) 3, 17, 39, 69, 107, ... un =
(b) 5, 18, 37, 62, 93, ... un =
(c) 9, 23, 45, 75, 113, ... un =
(d) –4, 12, 38, 74, 120, ... un =
Question 8

(a)

Calculate the second differences for the sequence,

9, 4, –5, –18, –35, ...

(b)

Determine the formula for the general term of the sequence.

un =
(c)

Calculate the 20th term of the sequence.

20th term = –2 × 202 + 20 + 10 = –800 + 30 = –770
Question 9

Determine the formula for the general term of the sequence,

6, 10, 12, 12, 10, 6, ...

un =
Question 10

(a)

Calculate the first, second and third differences for the sequence,

6, 13, 32, 69, 130, 221, ...

First differences: , , , ,

Second differences: , , ,

Third differences: , ,

(b)

Determine a formula for the general term of the sequence.

un =
Question 11

This is a series of patterns with grey and black tiles.

(a)

How many grey tiles and black tiles will there be in pattern number 8?

Grey tiles:

Black tiles:

(b)

How many grey tiles and black tiles will there be in pattern number 16?

Grey tiles:

Black tiles:

(c)

How many grey tiles and black tiles will there be in pattern number P?

Grey tiles:

Black tiles:

(d)

T = total number of grey tiles and black tiles in a pattern
P = pattern number

Use symbols to write down an equation connecting T and P.