Sequences, Patterns and Formulae
Sequences, Patterns and Formulae covers recognising and continuing number sequences, finding the nth term of a linear (arithmetic) sequence, describing the rule behind a growing diagram pattern such as a tiling or matchstick pattern, and substituting into and rearranging simple formulae.
Method
- For a sequence of numbers, find the term-to-term rule first by working out the difference between consecutive terms.
- If the difference between terms is always the same, the sequence is linear, and its nth term has the form (common difference x n) plus or minus an adjustment.
- To find the nth term of a linear sequence, multiply the common difference by n, then work out what must be added or subtracted so that n = 1 gives the first term.
- Recognise common special sequences by their pattern: square numbers (1, 4, 9, 16, ...), triangular numbers (1, 3, 6, 10, ...) and sequences where each term is the sum of the two before it.
- For a growing diagram pattern, count how many extra items are added each time the pattern grows by one step, and link this directly to the term-to-term rule and the nth term.
- To substitute into a formula, replace each letter with its given value carefully, keeping to the correct order of operations, especially with powers and brackets.
- To rearrange a formula to make a different letter the subject, apply the same operation to both sides of the equation until the required letter stands alone.
- Check an nth term rule by testing it against at least two terms of the original sequence, not just the first.
Worked example
A sequence begins 5, 9, 13, 17, ... Find an expression for the nth term, and use it to find the 20th term.
- Find the common difference between terms: 9 - 5 = 4 and 13 - 9 = 4, so the difference is always 4.
- Since the difference is constant, the nth term starts with 4n.
- Compare 4n with the sequence when n = 1: 4 x 1 = 4, but the first term is 5, so 1 must be added: nth term = 4n + 1.
- Check with the second term: 4 x 2 + 1 = 9, which matches. Substitute n = 20 to find the 20th term: 4 x 20 + 1 = 81. Final answer: nth term = 4n + 1, and the 20th term is 81.
Practice questions
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Q1Write the next two terms of the sequence: 3, 7, 11, 15, ...Show answer
Answer: 19, 23 (the term-to-term rule is add 4)
Q2Find the nth term of the sequence 2, 5, 8, 11, ...Show answer
Answer: 3n - 1 (common difference 3, and 3 x 1 - 1 = 2, which matches the first term)
Q3The nth term of a sequence is 5n - 2. Find the 12th term.Show answer
Answer: 58 (5 x 12 - 2 = 60 - 2 = 58)
Q4Write the first four terms of the sequence with nth term 2n^2.Show answer
Answer: 2, 8, 18, 32 (substituting n = 1, 2, 3, 4: 2 x 1, 2 x 4, 2 x 9, 2 x 16)
Q5A pattern of dots grows so that pattern 1 has 4 dots, pattern 2 has 7 dots, and pattern 3 has 10 dots. How many dots will pattern 6 have?Show answer
Answer: 19 dots (the rule is add 3 each time, giving nth term = 3n + 1, and 3 x 6 + 1 = 19)
Q6Using the formula v = u + at, find v when u = 5, a = 3 and t = 4.Show answer
Answer: 17 (v = 5 + 3 x 4 = 5 + 12 = 17)
Q7Rearrange the formula P = 2l + 2w to make w the subject.Show answer
Answer: w = (P - 2l) / 2
Q8The first three terms of the triangular numbers are 1, 3, 6. If the pattern continues in the same way, what is the next term after 6?Show answer
Answer: 10 (the differences between terms are 2 then 3, so the next difference is 4: 6 + 4 = 10)
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A sequence has nth term 6n - 4. (a) Find the 15th term. (b) Determine whether 158 is a term in this sequence, showing your reasoning.
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A tiling pattern is built from hexagonal tiles. Pattern 1 uses 6 tiles, pattern 2 uses 11 tiles, and pattern 3 uses 16 tiles. (a) Find an expression for the number of tiles in pattern n. (b) A pattern in this sequence uses 91 tiles. Find which pattern number this is.
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The cost, C pounds, of hiring a bouncy castle for h hours is given by the formula C = 45 + 12h. (a) Calculate the cost of hiring the bouncy castle for 5 hours. (b) A school has a budget of 120 pounds. Find the maximum whole number of hours they can hire the bouncy castle for, showing your working.
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Free printable worksheet
Want more practice on paper? Download the sequences, patterns and formulae worksheet pack - 10 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 15 of 13+ Maths Workbook, the whole course as one free printable PDF.
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