How to solve number series problems

Step one: write the differences

Under the series, write the difference between each pair of neighbours. For 7, 11, 15, 19 the differences are 4, 4, 4 and the next term is 23. For 2, 3, 5, 9, 17 the differences are 1, 2, 4, 8, which double, so the next difference is 16 and the next term is 33. If the differences are not constant, write the differences of the differences; a constant second difference means a quadratic rule such as squares.

Step two: test ratios

If differences grow quickly, divide neighbours instead. For 3, 6, 12, 24 each term is double the last, so the next is 48. For 81, 27, 9, 3 each term is a third of the last, so the next is 1. Ratios also expose mixed rules such as "double, then subtract two": 5, 10, 8, 16, 14, 28 alternates ×2 and −2, so the next term is 26.

Step three: look for alternation and recursion

Some series interleave two simple rules. In 10, 8, 9, 7, 8 the pattern is −2, +1, −2, +1, so the next term is 6. Others are recursive: in 1, 1, 2, 3, 5, 8 each term is the sum of the previous two. A quick test for recursion is to add or multiply the two previous terms and see whether the result matches the next one.

Step four: check the famous sequences

• Squares: 1, 4, 9, 16, 25, 36.
• Cubes: 1, 8, 27, 64, 125.
• Triangular numbers: 1, 3, 6, 10, 15, 21.
• Factorials: 1, 2, 6, 24, 120.
• Primes: 2, 3, 5, 7, 11, 13.

These appear often because they are easy to write and hard to guess without knowing them. Recognising them saves the arithmetic.

When a series has more than one answer

Any finite series can be continued by more than one rule; a well-written item makes one rule clearly simplest. If you find two rules that fit, choose the one with fewer steps, and note that in real tests the ambiguity is the item writer's error, not yours. Our numerical and pattern routes list the intended rule family for every item in the answer review, so you can see which mechanism the item was built from.