Unit 10 Section 2 : Finding the Formula for a Linear Sequence

It is possible to determine a formula for linear sequences, i.e. sequences where the difference between successive terms is always the same.

The first differences for the number pattern

111417202326...
are33333

If we look at the sequence 3n, i.e. the multiples of 3, and compare it with our original sequence

our sequence 11 14 17 20 23 26
sequence 3n 3 6 9 12 15 18

we can see easily that the formula that generates our number pattern is

nth term of sequence = 3n + 8
i.e. u_n = 3n + 8

If, however, we had started with the sequence

38 41 44 47 50 53 ...

the first differences would still have been 3 and the comparison of this sequence with the sequence 3n

our sequence 38 41 44 47 50 53
sequence 3n 3 6 9 12 15 18

would have led to the formula u_n = 3n + 35.

In the same way, the sequence

–7 –4 –1 2 5 8 ...

also has first differences 3 and the comparison

our sequence – 7 – 4 – 1 2 5 8
sequence 3n 3 6 9 12 15 18

yields the formula u_n = 3n – 10.

From these examples, we can see that any sequence with constant first difference 3 has the formula

u_n = 3n + c

where the adjustment constant c may be either positive or negative.

This approach can be applied to any linear sequence, giving us the general rule that:

If the first difference between successive terms is d, then
u_n = d × n + c

Example 1

Determine a formula for this sequence:

7,13,19,25,31,...

First consider the differences between the terms,

713192531...
6666

As the difference is always 6, we can write,

u_n = 6n + c

As the first term is 7, we can write down the equation:

7 = 6 × 1 + c
= 6 + c
c = 1

So the formula will be,

u_n = 6n + 1

We can check that this formula is correct by testing it on other terms, for example,

the 4th term = 6 × 4 + 1 = 25

which is correct.

Example 2

Determine a formula for this sequence:

2,7,12,17,22,27,...

First consider the differences between the terms,

2712172227...
55555

The difference between each term is always 5, so the formula will be,

u_n = 5n + c

The first term can be used to form an equation to determine c:

2 = 5 × 1 + c
2 = 5 + c
c = –3

So the formula will be,

u_n = 5n – 3

Note that the constant term, c, is given by

c = first term – first difference

Example 3

Determine a formula for the sequence:

28, 25, 22, 19, 16, 13, ...

First consider the differences between the terms,

282522191613...
–3–3–3–3–3

Here the difference is negative because the terms are becoming smaller.
Using the difference as –3 gives,

u_n = –3n + c

The first term is 28, so

28 = –3 × 1 + c
28 = –3 + c
c = 31

The general formula is then,

u_n = –3n + 31

or

u_n = 31 – 3n

Exercises

Question 1

For the sequence,

7, 11, 15, 19, ...

(a)

calculate the difference between successive terms,

(b)

determine the formula that generates the sequence.

un =
Question 2

Determine the formula for each of the following sequences:

(a)6, 10, 14, 20, 24, ... un =
(b)11, 13, 15, 17, 19, ... un =
(c)9, 16, 23, 30, 37, ... un =
(d)34, 56, 78, 100, 122, ... un =
(e)22, 31, 40, 49, 58, ... un =
Question 3

One number is missing from the following sequence:

1, 6, 11, , 21, 26, 31

(a)

Write in the missing number.

(b)

Calculate the difference between successive terms.

(c)

Determine the formula that generates the sequence.

un =
Question 4

Determine the general formula for each of the following sequences:

(a) 1, 4, 7, 10, 13, ... un =
(b) 2, 6, 10, 14, 18, ... un =
(c) 4, 13, 22, 31, 40, ... un =
(d) 5, 15, 25, 35, 45, ... un =
(e) 1, 20, 39, 58, 77, ... un =
Question 5

For the sequence,

18, 16, 14, 12, 10, ...

(a)

calculate the difference between successive terms,

(b)

determine the formula that generates the sequence.

un =
Question 6

Determine the general formula for each of the following sequences:

(a) 19, 16, 13, 10, 7, ... un =
(b) 100, 96, 92, 88, 84, ... un =
(c) 41, 34, 27, 20, 13, ... un =
(d) 66, 50, 34, 18, 2, ... un =
(e) 90, 81, 72, 63, 54, ... un =
Question 7

For the sequence,

–2, –4, –6, –8, –10, –12, ...

(a)

calculate the difference between successive terms,

(b)

determine the formula for the sequence.

un =
Question 8

Determine the formula that generates each of the following sequences:

(a) 0, – 5, – 10, – 15, – 20, ... un =
(b) – 18, – 16, – 14, – 12, – 10, ... un =
(c) – 5, – 8, – 11, – 14, – 17, ... un =
(d) 8, 1, – 6, – 13, – 20, ... un =
(e) – 7, – 3, 1, 5, 9, ... un =
Question 9

A sequence has first term 20 and the difference between the terms is always 31.

(a)

Determine a formula to generate the terms of the sequence.

(b)

Calculate the first 5 terms of the sequence.

, , , ,
Question 10

The second and third terms of a sequence are 16 and 27. The difference between successive terms in the sequence is always constant.

(a)

Determine the general formula for the sequence.

(b)

Calculate the first 5 terms of the sequence.

, , , ,
Question 11

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

The series of patterns continues by adding

each time.

(a)

Complete this table:

pattern numbernumber of
grey tiles
number of
white tiles
5
16
(b)

Complete this table by writing expressions:

pattern numberexpression for
the number of
grey tiles
expression for
the number of
white tiles
n
(c)

Write an expression to show the total number of tiles in pattern number n. Simplify your expression.

(d)

A different series of patterns is made with tiles.

The series of patterns continues by adding

each time.

For this series of patterns, write an expression to show the total number of tiles in pattern number n.
Simplify your expression.

The number of grey tiles follows the sequence 4, 6, 8, 10, ... with constant first difference 2 and nth term 2n + 2.

The number of white tiles follows the sequence 5, 8, 11, 14, ... with constant first difference 3, and nth term 3n + 2.

Total number of tiles = number of grey tiles + number of white tiles
= (2n + 2) + (3n + 2) = 5n + 4