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 nth term will include 2n, if the difference is 3 the expression for the nth 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 nth 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 nth 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 nth 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 nth term is –4n + 54 or 54 – 4n.

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

1st Differences          3    5     7     9

2nd Differences             2    2     2

The second differences are constant, 2. The expression for the nth term contains n2.

Compare the sequence to n2

Sequence              2    5   10   17   26

n2                             1    4     9   16   25

Each term in the sequence is 1 more than n2, so the expression for the nth term is n2 + 1