Module 6 (M6) – Number and algebra - Sequences

Each term in a sequence has a position. The first term is in position 1, the second term is in position 2 and so on.

Position to term rule is a formula which can be used to find any term in a sequence. This is also called the \(n_{th}\) term rule, in which \(n\) means any position in the sequence.

Example

Method 1

Work out the nth term of the following sequence: \(3, 5, 7, 9, …\)

Firstly, write out the sequence and the positions of the terms.

As there isn't a clear way of going from the position to the term, look for a common difference between the terms. In this case, there is a difference of 2 each time.

This common difference describes the times tables that the sequence is working in. In this sequence it's the 2 times tables.

Write out the 2 times tables and compare each term in the sequence to the 2 times tables.

To get from the position to the term, first multiply the positions by 2 then add 1.

If the position is \(n\), then this is \(2\times n+1\) which can be written as \(2n+1\).

Method 2

Working out the n^th rule

• Work out the common difference or term to term rule
  • This gives the coefficient of \(n\)
• Work backwards from the first term to find the number that should be added or subtracted to find each term.

Example

Work out the \(n_{th}\) term for the following sequence: \(3, 5, 7, 9, …\)

Solution

• Work out the common difference or term to term rule.

This gives the coefficient of \(n\), in this case 2.

The \(n_{th}\) term begins \(2n\).

• Work backwards from the first term to find the number that should be added or subtracted to find each term.

The \(n_th\) term for the sequence is \(2n – 2\).

If the nth term of a sequence is known, it is possible to work out any number in that sequence.

Example

Write the first five terms of the sequence \(3n+4\).

\(n\) represents the position in the sequence. The first term in the sequence is when \(n=1\), the second term in the sequence is when \(n=2\), and so on.

To find the terms, substitute \(n\) for the position number:

• when \(n=1,3n+4=3\times1+4=3+4=7\)
• when \(n=2,3n+4=3\times2+4=6+4=10\)
• when \(n=3,3n+4=3\times3+4=9+4=13\)
• when \(n=4,3n+4=3\times4+4=12+4=16\)
• when \(n=5,3n+4=3\times5+4=15+4=19\)

The first five terms of the sequence: \(3n+4\) are \(7, 10, 13, 16, 19, …\)