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.
Term to term rules
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 in the following sequence and write down the term to term rule:
3, 7, 11, 15, ...
Firstly, work out the 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'.
Question
What are the next 3 terms of a sequence that has a first term of 1, where the term to term rule is multiply by 2?
As an enthusiast and expert in mathematical sequences, I bring a wealth of knowledge and experience to elucidate the concepts discussed in the article. My proficiency is demonstrated through a deep understanding of various types of number sequences and the associated rules governing their progression.
Let's delve into the key concepts mentioned in the article:
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Number Sequences:
- Number sequences are sets of numbers that follow a specific pattern or rule.
- Each individual number in a sequence is referred to as a term.
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Special Sequences:
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Square Numbers: These are numbers that result from squaring the natural numbers. The nth term is given by (n^2). Example: 1, 4, 9, 16, 25, 36, ...
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Cube Numbers: Numbers obtained by cubing the natural numbers. The nth term is (n^3). Example: 1, 8, 27, 64, 125, ...
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Triangular Numbers: These numbers form a triangle of dots, and the term-to-term rule involves adding one more each time. Example: 1, 3, 6, 10, 15, ...
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Fibonacci Sequence: A sequence where each term is the sum of the two preceding terms. Example: 1, 1, 2, 3, 5, 8, 13, ...
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Linear Sequences:
- A sequence that increases or decreases by the same amount each time is termed a linear sequence.
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Term-to-Term Rules:
- Describes how to move from one term to the next in a sequence.
- In linear sequences, the rule can involve addition or subtraction.
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Examples from the Article:
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Example 1: The sequence 3, 7, 11, 15, ... increases by 4 each time. The term-to-term rule is 'add 4.'
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Example 2: The sequence -1, -0.5, 0, 0.5, ... has a term-to-term rule of 'add 0.5.'
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Questions:
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Question 1: For the sequence 17, 14, 11, 8, ..., the term-to-term rule is 'subtract 3,' and the next two terms are 5 and 2.
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Question 2: For a sequence starting with 1 and a term-to-term rule of 'multiply by 2,' the next three terms would be 2, 4, and 8.
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In conclusion, a comprehensive understanding of number sequences involves recognizing various types of sequences, understanding term-to-term rules, and applying these concepts to determine the next terms in a sequence. The examples provided illustrate the practical application of these principles.
