Module 5 (M5) – Algebra - Sequences

Number sequences are sets of numbers that follow a pattern or a rule.

Each number in a sequence is called a term.

There are some special sequences that you should recognise.

The most important of these are:

• square numbers: \(1, 4, 9, 16, 25, 36, …\) - the nth term is \(n^2\)
• cube numbers: \(1, 8, 27, 64, 125, …\) - the nth term is \(n^3\)
• triangular numbers: \(1, 3, 6, 10, 15, …\) (these numbers can be represented as a triangle of dots). The term to term rule for the triangle numbers is to add one more each time: \(1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10\) etc.
• Fibonacci sequence: \(1, 1, 2, 3, 5, 8, 13, …\) (in this sequence you start off with 1 and then to get each term you add the two terms that come before it)
• A sequence which increases or decreases by the same amount each time is called a linear sequence e.g.

The term to term rule of a sequence describes how to get from one term to the next.

Example 1

Work out the next two terms in the following sequence and write down the term to term rule:

\(3, 7, 11, 15, …\)

Firstly, work out the difference in the terms.

This sequence is going up by four each time, so add 4 on to the last term to find the next term in the sequence.

\(3, 7, 11, 15, 19, 23, …\)

To work out the term to term rule, give the starting number of the sequence and then describe the pattern of the numbers.

The first number is 3. The term to term rule is 'add 4'.

Once the first term and term to term rule are known, all the terms in the sequence can be found.

Example 2

Work out the next two terms in the following sequence and write down the term to term rule:

\(–1, -0.5, 0, 0.5, …\)

The first term is –1. The term to term rule is 'add 0.5'.

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 rules use algebra to work out what number is in a sequence if the position in the sequence is known. This is also called the nth term, which is a position to term rule that works out a term at position \(n\), where \(n\) means any position in the sequence.

Example

Work out the position to term rule for the following sequence: \(5, 6, 7, 8, …\)

Solution

First, write out the sequence and the positions of each term.

Next, work out how to go from the position to the term.

In this example, to get from the position to the term, take the position number and add 4.

If the position is \(n\), then the position to term rule is \(n+4\).

The nth term of a sequence is the position to term rule using \(n\) to represent the position number.

Example

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 \(2n+1\).

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 \(7, 10, 13, 16, 19, …\)

The \(n^{th}\) for a quadratic sequence has a term that contains \(x^2\). Terms of a quadratic sequence can be worked out in the same way.

Example

Write the first five terms of the sequence \(n^2+3n–5\).

• when \(n=1, n^2 + 3n–5 = 1^2 + 3 \times 1–5 = 1 + 3–5 = –1\)
• when \(n=2, n^2 + 3n–5 = 2^2 + 3 \times 2–5 = 4 + 6–5 = 5\)
• when \(n=3, n^2 + 3n–5 = 3^2 + 3 \times 3–5 = 9 + 9–5 = 13\)
• when \(n=4, n^2 + 3n–5 = 4^2 + 3 \times 4–5 = 16 + 12–5 = 23\)
• when \(n=5, n^2 + 3n–5 = 5^2 + 3 \times 5–5 = 25 + 15–5 = 35\)

The first five terms of the sequence: \(n^2 + 3n – 5\) are \(–1, 5, 13, 23, 35\).

Some sequences may contain terms that are fractions.

Example

Work out the \(nth\) term of the following sequence: \(\frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{4}{9}…\)

First, look for the pattern on the numerator of each fraction.

The rule is \(n\).

Then, look for the pattern on the denominator of each equation:

The rule is \(2n+1\)

The rule for the sequence \(\frac{1}{3},\frac{2}{5},\frac{3}{7},\frac{4}{9}…\) is \(\frac{n}{2n+1}\)