Algebra
Sequences: Term-to-Term Rules and the nth Term
Year 7 · Year 8 · Year 9
- ✓Identify and describe term-to-term rules for sequences.
- ✓Generate terms of a sequence given a term-to-term rule and a starting term.
- ✓Recognise linear sequences by their constant difference.
- ✓Find the nth term expression for a linear sequence.
- ✓Use the nth term to find any term in a linear sequence.
Key concepts
A sequence is an ordered list of numbers, shapes, or other mathematical objects, usually following a particular pattern or rule. Each number in a sequence is called a 'term'. For example, in the sequence 2, 4, 6, 8, ..., 2 is the 1st term, 4 is the 2nd term, and so on.
A term-to-term rule is a rule that tells you how to get from one term in a sequence to the next term. It often involves a consistent operation such as adding a number, subtracting a number, multiplying by a number, or dividing by a number. For example, in the sequence 3, 6, 9, 12, ..., the term-to-term rule is 'add 3'.
A linear sequence (also known as an arithmetic sequence) is a sequence where the difference between consecutive terms is constant. This constant difference is called the 'common difference'. For example, 5, 8, 11, 14, ... is a linear sequence with a common difference of 3.
The nth term is a formula or algebraic expression that allows you to find any term in a sequence directly, without having to list all the previous terms. For a linear sequence, the nth term is always in the form 'an + b', where 'a' is the common difference and 'b' is a constant.
Key facts to remember
- 1A sequence is an ordered list of numbers following a rule.
- 2Each number in a sequence is called a 'term'.
- 3A term-to-term rule describes how to get from one term to the next.
- 4A linear sequence has a constant difference between consecutive terms.
- 5The nth term of a linear sequence is always in the form 'an + b'.
- 6In the nth term 'an + b', 'a' represents the common difference of the sequence.
- 7To find 'b' in 'an + b', subtract the common difference ('a') from the first term of the sequence.
- 8The nth term allows you to find any term in the sequence directly, without listing all previous terms.
Worked examples
Example 1
Find the next three terms of the sequence: 4, 9, 14, 19, ... given the term-to-term rule is 'add 5'.
Answer
24, 29, 34
Always ensure you apply the rule consistently to find the subsequent terms.
Example 2
For the sequence 15, 12, 9, 6, ... find the term-to-term rule and the 8th term.
Answer
Term-to-term rule: Subtract 3. 8th term: -6.
Be careful when working with negative numbers and subtraction.
Example 3
Find the nth term of the linear sequence: 7, 11, 15, 19, ...
Answer
4n + 3
The 'a' in 'an + b' is always the common difference. The 'b' is the value of the '0th term'.
Common mistakes
- ✗Confusing the term number (n) with the actual value of the term.
- ✗Incorrectly calculating the common difference, especially when negative numbers are involved.
- ✗Forgetting to find the 'b' part of the 'an + b' rule, or calculating it incorrectly.
- ✗Not checking the derived nth term rule by substituting values of 'n' to see if it generates the correct terms.
- ✗Applying the term-to-term rule incorrectly or inconsistently when generating terms.
Exam tips
- ★Always show your working clearly, especially when finding the common difference and deriving the nth term expression.
- ★Check your nth term rule by substituting n=1 and n=2 to ensure it generates the first two terms of the given sequence.
- ★Read the question carefully to determine if it asks for a term-to-term rule, a specific term, or the general nth term.
- ★If a sequence involves negative numbers, take extra care when calculating the differences between terms.
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