A22 finding the nth term

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A22 Finding the nth term

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Sequences of multiples

All sequences of multiples can be generated by adding the same amount each time. They are linear sequences.

For example, the sequence of multiples of 5:

5, 10, 15, 20, 25, 30, 35, 40, …

+5 +5 +5 +5 +5 +5 +5

can be found by adding 5 each time.

Compare the terms in the sequence of multiples of 5 to their position in the sequence:

Position

Term

1

5

2

10

3

15

4

20

5

25

n…

…

× 5 × 5 × 5 × 5 × 5 × 5

5n



The sequence of multiples of 3:3, 6, 9, 12, 15, 18, 21, 24, …

+3 +3 +3 +3 +3 +3 +3


Compare the terms in the sequence of multiples of 3 to their position in the sequence:

Position

Term

1

3

2

6

3

9

4

12

5

15

n…

…×3 ×3 ×3 ×3 ×3 ×3

3n

The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.



The nth term of a sequence of multiples is always dn, where d is the difference between consecutive terms.

For example:

The nth term of 4, 8, 12, 16, 20, 24, … is 4n

The 10th term of this sequence is 4 × 10 = 40




Finding the nth term of a linear sequence

The terms in this sequence4, 7, 10, 13, 16, 19, 22, 25, …

+3 +3 +3 +3 +3 +3 +3


Compare the terms in the sequence to the multiples of 3.

Position

Multiples of 3

1 2 3 4 5 n…× 3 × 3 × 3 × 3 × 3 × 3

3n

Term 4 7 10 13 16 …

3 6 9 12 15+ 1 + 1 + 1 + 1 + 1 + 1

3n + 1

Each term is one more than a multiple of 3.



The terms in this sequence1, 6, 11, 16, 21, 26, 31, 36, …

+5 +5 +5 +5 +5 +5 +5


Compare the terms in the sequence to the multiples of 5.

Position

Multiples of 5

1 2 3 4 5 n…× 5 × 5 × 5 × 5 × 5 × 5

5n

Term 1 6 11 16 21 …

5 10 15 20 25– 4 – 4 – 4 – 4 – 4 – 4

5n – 4

Each term is four less than a multiple of 5.



The terms in this sequence5, 3, 1, –1, –3, –5, –7, –9, …

–2 –2 –2 –2 –2 –2 –2

can be found by subtracting 2 each time.

Compare the terms in the sequence to the multiples of –2.

Position

Multiples of –2

1 2 3 4 5 n…× –2 × –2 × –2 × –2 × –2 × –2

–2n

Term 5 3 1 –1 –3 …

–2 –4 –6 –8 –10+ 7 + 7 + 7 + 7 + 7 + 7

7 – 2n

Each term is seven more than a multiple of –2.


Sequences that increase (or decrease) in equal steps are called linear or arithmetic sequences.

The difference between any two consecutive terms in an arithmetic sequence is a constant number.

When we describe arithmetic sequences we call the difference between consecutive terms, d.

We call the first term in an arithmetic sequence, a.

If an arithmetic sequence has a = 5 and d = –2,

We have the sequence:

5, 3, 1, –1, –3, –5, ...

Arithmetic sequences


The rule for the nth term of any arithmetic sequence is of the form: T(n) = an + b

a and b can be any number, including fractions and negative numbers.For example:

T(n) = 2n + 1 Generates odd numbers starting at 3.

T(n) = 2n + 4 Generates even numbers starting at 6.

T(n) = 2n – 4 Generates even numbers starting at –2.

T(n) = 3n + 6 Generates multiples of 3 starting at 9.

T(n) = 4 – n Generates descending integers starting at 3.

The nth term of an arithmetic sequence

A22 finding the nth term

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