**Sequences**

**Sequence ** – a list of numbers in a given order

**Terms** – the numbers in a sequence

**Consecutive terms** – terms next to one another in a sequence

A sequence is a pattern of numbers, connected by a rule.

**Arithmetic sequences
**

In arithmetic sequences, the difference between consecutive terms is a constant or fixed number. Arithmetic sequences are sometimes referred to as linear sequences.

For example, the sequence 3, 5, 7, 9, … is an arithmetic sequence with a constant difference of 2 between consecutive terms.

** Finding the nth term of an arithmetic sequence**

Find the difference between consecutive terms.

If the difference is 2 then your expression for the *n*th term will include 2n, if the difference is 3 the expression for the *n*th term will include 3n

Compare the sequence with the values of 2n or difference x *n* to find what you need to add or subtract to give your final expression for the *n*th term.

Using the example above:

Sequence 3 5 7 9 11

2n 2 4 6 8 10

Each term in the sequence is 1 more than 2n, so the expression for the *n*th term is **2n + 1**

What if the numbers in the sequence are going down instead of up?

The same principles apply.

For example, take the sequence 50, 46, 42, 38, 34, …

The difference between consecutive terms is –4, so the expression for the *n*th term contains –4n

Sequence 50 46 42 38 34

-4n -4 -8 -12 -16 -20

Each term in the sequence is 54 more than –4n, so the expression for the *n*th term is **–4n + 54** or **54 – 4n**.

** Quadratic sequences**

In a quadratic sequence, the difference between consecutive terms is not constant.

For example, the sequence 2, 5, 10, 17, 26, … is a quadratic sequence. The first differences are not constant, 3, 5, 7, 9, … so, now it’s time to look at the second differences. Second differences are the differences between consecutive (first) differences.

Sequence 2 5 10 17 26

1^{st} Differences 3 5 7 9

2^{nd} Differences 2 2 2

The second differences are constant, 2. The expression for the *n*th term contains *n ^{2}*.

Compare the sequence to *n ^{2}*

Sequence 2 5 10 17 26

*n ^{2}* 1 4 9 16 25

Each term in the sequence is 1 more than *n ^{2}, *so the expression for the

*n*term is

^{th}

*n*+ 1^{2}

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