Algebraic expressions & sequences · Ages 12–13
Patterns in sequences
Look at neighbours: same gap, same multiple, or the sum of the two before.
How to do it
Method
Look at neighbours: same gap, same multiple, or the sum of the two before
3, 7, 11, 15: add 4 each time, next 15 + 4 = 19
Another way
Write out the gaps
1, 3, 6, 10: gaps 2, 3, 4, next 10 + 5 = 15
In full
A sequence is a list of numbers following a rule, each one a term. Look at neighbours first: the same gap makes an arithmetic sequence (3, 7, 11, 15, adding 4); the same multiple makes a geometric one (2, 6, 18, 54, times 3); each term the sum of the two before makes a Fibonacci sequence (1, 1, 2, 3, 5, 8); a gap that keeps growing may mean polygon numbers (triangle numbers 1, 3, 6, 10, adding one more each time; square numbers 1, 4, 9, 16). Knowing the rule, you can go forwards; to go backwards, do the opposite: take off the common difference, divide by the common ratio, or for Fibonacci take the earlier term from the later one. Or write the gaps in a row beneath, where the pattern is plain to see. The answer is the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
An arithmetic sequence: 10, −2, −14, −26,
Show the answer and working
Answer: −38
Each step is −12: −26 − 12 = −38.
Example 2
An arithmetic sequence, going backwards: , 17, 28, 39
Show the answer and working
Answer: 6
Going forwards adds 11, so going backwards takes it off: 17 − 11 = 6.
Example 3
Which of these is a geometric sequence?
- A. 2, 6, 10, 14
- B. 2, 6, 18, 54
- C. 2, 6, 14, 26
- D. 2, 6, 8, 14
Choose the right answer
Show the answer and working
Answer: B. 2, 6, 18, 54
A geometric sequence multiplies by the same number: 2 × 3 = 6, 6 × 3 = 18, 18 × 3 = 54.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.An arithmetic sequence: 19, 30, 41, 52,
Answer:
2.A geometric sequence: 5, 15, 45, 135,
Answer:
3.An arithmetic sequence, going backwards: , 29, 32, 35
Answer:
4.Which of these is a geometric sequence?
- A. 2, 6, 14, 26
- B. 2, 6, 8, 14
- C. 2, 6, 10, 14
- D. 2, 6, 18, 54
Choose the right answer
Answer:
5.The sequence's rule: each term = the term before × 2 − 13. Term 1 is 10. Term 4 is .
Answer:
All the answers
- 1. 63
- 2. 405
- 3. 26
- 4. D. 2, 6, 18, 54
- 5. −11
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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