Arithmetic Sequences

If the term-to-term rule for a sequence is to add or subtract the same number each time, it is called an arithmetic sequence, eg:

4, 9, 14, 19, 24, ... or 8, 7.5, 7, 6.5, 6, ...

The position-to-term rule (or the \(nth\) term) of an arithmetic sequence is of the form \(an + b\). eg:

\(5n − 1\) or \(-0.5n + 8.5\) are the position-to-term rules for the two examples above.

Arithmetic sequences are also known as linear sequences because, if you plot the position on a horizontal axis and the term on the vertical axis, you get a linear (straight line) graph.

Quadratic sequences

The \(nth\) term of a quadratic sequence has the form \(an^2 + bn + c\). (Notice how this is the same form as used for quadratic equations.) Any term of the quadratic sequence can be found by substituting for \(n\), like before.

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

Working out position-to-term rules for arithmetic sequences

Example

Work out the position to term rule for the following sequence: 5, 6, 7, 8, ...

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

Position	1	2	3	4
Term	5	6	7	8

Position
1
2
3
4

Term
5
6
7
8

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

Position	1	2	3	4
Operation	\(+4\)	\(+4\)	\(+4\)	\(+4\)
Term	5	6	7	8

Position
1
2
3
4

Operation
\(+4\)
\(+4\)
\(+4\)
\(+4\)

Term
5
6
7
8

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

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

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.

Number positions of a sequence with +2 increments

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 gives the times table used in the sequence and the first part of the position-to-term rule. In this sequence it's the 2 times tables.

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

Position	1	2	3	4
Operation	\(\times 2\)	\(\times 2\)	\(\times 2\)	\(\times 2\)
2 times table	2	4	6	8
Operation	\(+ 1\)	\(+ 1\)	\(+ 1\)	\(+ 1\)
Term	3	5	7	9

Position
1
2
3
4

Operation
\(\times 2\)
\(\times 2\)
\(\times 2\)
\(\times 2\)

2 times table
2
4
6
8

Operation
\(+ 1\)
\(+ 1\)
\(+ 1\)
\(+ 1\)

Term
3
5
7
9

To get from the position to the term, first multiply the position by 2 then add 1. If the position is \(n\), then this is \(2 \times n + 1\) which can be written as \(2n + 1\).

Geometric sequences

If the term-to-term rule for a sequence is to multiply or divide by the same number each time, it is called a geometric sequence, eg 3, 9, 27, 81, 243, ...

Example

Work out the first three terms of \((\frac{3}{4})^n\)

\((\frac{3}{4})^1\) = \(\frac{3}{4}\)

\((\frac{3}{4})^2\) = \(\frac{9}{16}\)

\((\frac{3}{4})^3\) = \(\frac{27}{64}\)

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Sequences - OCRArithmetic Sequences