Find a Determining Formula an for the Following Sequence Assuming the Pattern Continues

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

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,

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,

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,

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.

u_n =

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.

u_n =

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.

u_n =

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.

u_n =

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 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 number	number of grey tiles	number of white tiles
5
16

(b)

Complete this table by writing expressions:

pattern number	expression 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

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Source: https://www.cimt.org.uk/projects/mepres/book9/bk9i10/bk9_10i2.html

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