Number Series: Patterns and Methods
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Key points
- Write the differences first, then the differences of those differences
- Fast growth points to a ratio or to a form like x2 + 1
- Rising and falling terms point to an alternate series
- In a wrong-number question the faulty term sits between two abnormal differences
- Memorise squares to 30, cubes to 15 and the 25 primes below 100
- 1 is not prime and 2 is the only even prime
What a number series is
A number series is a row of numbers arranged by a fixed rule. Questions come in two forms: find the missing number, or identify the wrong number.
The main patterns
| Pattern | Clue in the series | Method |
|---|---|---|
| Arithmetic progression | Differences are constant | nth term = a + (n − 1) x d |
| Difference of differences | The differences themselves form a series | Continue the difference series |
| Geometric progression | Numbers grow by a constant ratio | Divide each term by the previous one |
| Mixed operation | Fast growth with a ratio close to 2 or 3 | Test x2 + 1, x2 − 1, x3 + 1 and so on |
| x n + n | Growth accelerates sharply | Multiplier and addend both rise 1, 2, 3 |
| Squares or cubes | Terms sit near 4, 9, 16 or near 8, 27, 64 | Compare each term with n squared or n cubed |
| Prime based | Terms are primes or squares of primes | Recall the prime list |
| Alternate series | Terms rise and fall | Split the odd and even positions |
| Fibonacci type | Each term equals the sum of the previous 2 or 3 | Add the previous terms |
How to attack a series
- Write the difference between consecutive terms.
- If the differences are not constant, look at the differences of those differences.
- If the numbers grow fast, check the ratio or a form such as x a + b.
- If the terms rise and fall, suspect an alternate series and split the positions.
- If the terms sit near familiar squares, cubes or primes, test that pattern.
- Apply the rule you found to every term before marking the answer.
Key point
In a wrong-number question, the faulty term spoils the difference on both sides of it. The term sitting between two abnormal differences is the wrong one.
Values worth memorising
- Squares 11 to 30: 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900
- Cubes 1 to 15: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375
- Primes up to 100, 25 in all: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Exam tip
The number 1 is not prime, and 2 is the only even prime. Both points appear regularly in counting questions.
Practice questions
Answer all, then check. Explanations appear after checking.
Finished this topic? Tick it off.
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