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A10 Generating sequences

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Sequence grid

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Predicting terms in a sequence

Usually, we can predict how a sequence will continue by looking for patterns.

For example: 87, 84, 81, 78, ...

We can predict that this sequence continues by subtracting 3 each time.

However, sequences do not always continue as we would expect.

For example:

A sequence starts with the numbers 1, 2, 4, ...

How could this sequence continue?

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Writing sequences from term-to-term-rules

A term-to-term rule gives a rule for finding each term of a sequence from the previous term or terms.

To generate a sequence from a term-to-term rule we must also be given the first number in the sequence.

For example:

1st term5

Term-to-term rule

Add consecutive even numbers starting with 2.This gives us the sequence,

5

+2

7

+4

11

+6

17

+8

25

+10

35

+12

47 ...

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2nd1st

3Term

Sometimes sequences are arranged in a table like this:

nth…6th5th4th3rdPosition

Sequences from position-to-term rules

We can say that each term can be found by multiplying the position of the term by 3.

This is called a position-to-term rule.

For this sequence we can say that the nth term is 3n, where n is a term’s position in the sequence.

What is the 100th term in this sequence?

3 × 100 = 300

3n…18151296

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Sequences from position-to-term rules

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Writing sequences from position-to-term rules

The position-to-term rule for a sequence is very useful because it allows us to work out any term in the sequence without having to work out any other terms.

We can use algebraic shorthand to do this.

We call the first term T(1), for Term number 1,

we call the second term T(2),

we call the third term T(3), ...

and we call the nth term T(n).

T(n) is called the the nth term or the general term.

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For example, suppose the nth term of a sequence is 4n + 1.

We can write this rule as: T(n) = 4n + 1

Find the first 5 terms.

T(1) = 4 × 1 + 1 = 5

T(2) = 4 × 2 + 1 = 9

T(3) = 4 × 3 + 1 = 13

T(4) = 4 × 4 + 1 = 17

T(5) = 4 × 5 + 1 = 21

The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.

Writing sequences from position-to-term rules