Sequences and Patterns
Sequences and patterns are lists of numbers or shapes that follow a rule linking each term to the next, such as adding a fixed amount, or a rule linking each term to its position, called the nth term. 11+ maths papers test finding missing terms, describing the rule in words, working out the nth term formula, and checking whether a given number belongs to the sequence.
Method
- Look at the differences between consecutive terms; if the difference is always the same, it is a linear sequence.
- If the differences are not constant, check whether the differences themselves form a pattern, or whether each term is found by doubling, or by adding the two previous terms.
- To find the nth term of a linear sequence, multiply n by the common difference, then adjust the constant so the formula gives the correct first term.
- Use the nth-term formula to jump straight to any term, such as the 20th term, without listing every term in between.
- To check whether a given number is in the sequence, set the nth-term formula equal to that number and solve for n; if n is a positive whole number, the value is in the sequence.
- For pattern (picture) sequences, count how many extra items are added each time to find the term-to-term rule, and link this to the algebraic nth term.
Worked example
A sequence begins 6, 11, 16, 21, ... (a) Find the nth term of the sequence. (b) Use your formula to find the 30th term. (c) Determine whether 150 is a term in the sequence.
- Find the common difference between consecutive terms: 11 - 6 = 5 and 16 - 11 = 5, so the sequence increases by 5 each time.
- The nth term has the form 5n + c. Using the first term (n = 1): 5(1) + c = 6, so c = 1. The nth term is 5n + 1.
- Substitute n = 30 into the formula: 5(30) + 1 = 151. This is the answer to part (b).
- For part (c), set 5n + 1 = 150 and solve: 5n = 149, so n = 29.8.
- Since n = 29.8 is not a whole number, 150 is not a term in the sequence.
- (a) nth term = 5n + 1. (b) 30th term = 151. (c) 150 is not a term, since n = 29.8 is not a whole number.
Practice questions
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Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
A sequence begins 50, 43, 36, 29, ... Write down the next two terms and describe the term-to-term rule.
A sequence has nth term 6n - 4. (a) Work out the 15th term. (b) Determine whether 68 is a term in this sequence.
A pattern of tiles is arranged so that pattern 1 has 3 tiles, pattern 2 has 7 tiles, pattern 3 has 11 tiles and pattern 4 has 15 tiles. (a) Find an expression, in terms of n, for the number of tiles in pattern n. (b) Use your expression to find the number of tiles in pattern 25. (c) A pattern in the sequence has 75 tiles. Find which pattern number this is.
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