Unit 10 Section 1 : Constant Differences

In the first part of this unit we consider sequences where the difference between successive terms is the same every time. We also use formulae to create the terms of a sequence.

Example 1

Write down the next 3 terms of each of the following sequences:

(a)

7, 11, 15, 19, 23, ...

711151923...
4444
The difference between each term and the next is always 4. This value is called the first difference. So we can continue the sequence by adding 4 each time. This gives the sequence:

7, 11, 15, 19, 23, 27, 31, 35
(b)

1, 9, 17, 25, 33, ...

19172533...
8888
Here the difference between each term and the next is always 8. To continue the sequence we must keep on adding 8 every time. This gives the sequence:

1, 9, 17, 25, 33, 41, 49, 57

Example 2

A sequence is defined by the formula u_n = 3n + 1.
Calculate the first 5 terms of this sequence.

The first term, often called u_1, is formed by substituting n = 1 into the formula.

u_1 = 3 × 1 + 1 = 3 + 1 = 4

For the second term, substitute n = 2 to give:

u_2 = 3 × 2 + 1 = 7

For the third term, substitute n = 3 to give:

u_3 = 3 × 3 + 1 = 10

For the fourth term, substitute n = 4 to give:

u_4 = 3 × 4 + 1 = 13

For the fifth term, substitute n = 5 to give:

u_5 = 3 × 5 + 1 = 16

So the first 5 terms of the sequence are

4, 7, 10, 13, 16.

Example 3

The terms of a sequence are given by the formula u_n = 8n – 3.
Calculate:

(a)

the first 3 terms of the sequence,

n = 1 gives u_1 = 8 × 1 – 3 = 5

n = 2 gives u_2 = 8 × 2 – 3 = 13

n = 3 gives u_3 = 8 × 3 – 3 = 21

So the first 3 terms are

5, 13, 21.

(b)

the 100th term of the sequence,

n = 100 gives u_100 = 8 × 100 – 3 = 797

So the 100th term of the sequence is 797.

(c)

the 200th term of the sequence.

n = 200 gives u_200 = 8 × 200 – 3 = 1597

So the 200th term of the sequence is 1597.

Exercises

Question 1

Write down the next 3 terms of each of the following sequences:

(a)2, 5, 8, 11, 14, ... , ,
(b)9, 18, 27, 36, 45, ... , ,
(c)13, 14, 15, 16, 17, ... , ,
(d)7, 15, 23, 31, 39, ... , ,
Question 2

Write down the next 3 terms of each of the following sequences:

(a)100, 98, 96, 94, 92, ... , ,
(b)20, 17, 14, 11, 8, ... , ,
(c)48, 43, 38, 33, 28, ... , ,
(d)17, 13, 9, 5, 1, ... , ,
Question 3

A sequence is defined by the formula u_n = 6n – 2.

(a)

Calculate the first 5 terms of the sequence.

, , , ,
(b)

What is the difference between the terms of the sequence?

Question 4

A sequence is defined by the formula u_n = 8n + 2.

(a)

Calculate the first 5 terms of the sequence.

, , , ,
(b)

What is the difference between the terms of the sequence?

(c)

Write down the next 3 terms of the sequence.

, ,
Question 5

A sequence is given by u_n = 7n – 3.

(a)

Calculate the first 4 terms of the sequence.

, , ,
(b)

What is the difference between the terms of the sequence?

The first difference 7 is the multiple of n in the formula u_n = 7n – 3.
Question 6

A sequence is given by u_n = 9n + 2.

(a)

Calculate the first 4 terms of the sequence.

, , ,
The first difference 9 is the multiple of n in the formula u_n = 9n + 2.
Question 7

A sequence is given by the formula u_n = 11n – 7.

(a)

What would you expect to be the difference between the terms of the sequence?

(b)

Calculate the first 4 terms of the sequence and check your answer to part (a).

, , ,
(c)

Calculate the 10th term of the sequence.

Question 8

A sequence is defined by the formula u_n = 82 – 4n.

(a)

Calculate the first 5 terms of the sequence.

, , , ,
(b)

What is the difference between terms for the sequence?

The first difference –4 is the multiple of n in the formula u_n = 82 – 4n.
(c)

Calculate the 20th term of the sequence.

Question 9

(a)

Calculate the 100th term of the sequence given by u_n = 8n – 5.

(b)

Calculate the 25th term of the sequence given by u_n = 11n – 3.

(c)

Calculate the 200th term of the sequence given by u_n = 3n + 22.

(d)

Calculate the 58th term of the sequence defined by u_n = 1000 – 5n.

Question 10

Four sequences, A, B, C and D, are defined by the following formulae:

Au_n = 8n + 2
Bu_n = 7n – 3
Cu_n = 3n + 1
Du_n = 100 – 6n
(a)

Which sequences have 4 as their first term?

(b)

Which sequence is decreasing?

(c)

Which sequence has a difference of 7 between terms?

(d)

Which sequence has 301 as its 100th term?

Question 11

(a)

Look at this part of a number line. Write down the 2 missing numbers.

Complete this sentence:

The numbers on this line go up in steps of .

(b)

This is a different number line. Write down the 3 missing numbers.

Complete this sentence:

The numbers on this line go up in steps of .

Question 12

Jeff makes a sequence of patterns with black and grey triangular tiles.

The rule for finding the number of tiles in pattern number N in Jeff's sequence is:

number of tiles = 1 + 3N
(a)

The 1 in this rule represents the black tile.
What does the 3N represent?

in pattern N

(b)

Jeff makes pattern number 12 in his sequence.
How many black tiles and how many grey tiles does he use?

black:
grey:
(c)

Jeff uses 61 tiles altogether to make a pattern in his sequence.
What is the number of the pattern he makes?

(d)

Barbara makes a sequence of patterns with hexagonal tiles.

Each pattern in Barbara's sequence has 1 black tile in the middle. Each new pattern has 6 more grey tiles than the pattern before.
Complete the rule for finding the number of tiles in pattern number N in Barbara's sequence.

number of tiles = +
Question 13

Owen has some tiles like these:

He uses the tiles to make a series of patterns.

(a)

Each new pattern has more tiles than the one before.
The number of tiles goes up by the same amount each time.

How many more tiles does Owen add each time he makes a new pattern?

tiles

(b)

How many tiles will Owen need altogether to make pattern number 6 ?

tiles
(c)

How many tiles will Owen need altogether to make pattern number 9 ?

tiles
(d)

Owen uses 40 tiles to make a pattern.

What is the number of the pattern he makes?

pattern number