2, 6, 12, 20, 30, ?
💡 Hint
Look at the gaps between neighboring terms, then at how those gaps change. Also compare each term with the position it holds in the list.
Reveal answer
42
Step-by-step solution
- Differences: 6 – 2 = 4, 12 – 6 = 6, 20 – 12 = 8, 30 – 20 = 10. The gaps are 4, 6, 8, 10.
- The gaps grow by 2 each time, so the next gap is 12, and 30 + 12 = 42.
- Second method: each term is the product of two consecutive whole numbers: 1×2 = 2, 2×3 = 6, 3×4 = 12, 4×5 = 20, 5×6 = 30.
- The next product is 6×7 = 42. Both methods agree.
- In formula form, term number n is n(n + 1). Neighboring terms differ by 2n + 2, which is why the gaps climb by exactly 2.
Why it fools people
Sequences fool people when they try to see the whole rule at once. The jumps here grow, so the eye looks for something dramatic like doubling or squaring. Squares come close (1, 4, 9, 16, 25), since each term is a square plus a little extra, but they never match. The steady, quiet pattern lives in the gaps, and you only see it once you write them down. Recording differences turns a guessing game into a simple, repeatable routine.
Did you get it?
About This Puzzle
Number sequences are a staple of brain training, aptitude tests and puzzle books for a good reason: every sequence hides a rule, and finding it is a small act of detective work. This one looks gentle, but it’s a fine test of method, because the rule can be found in two different ways, one by studying the gaps and one by studying the numbers themselves. When two independent methods give the same answer, you can be confident you’ve found the real rule and not a coincidence. Try to find both before you check.
How to Think About It
Start with the gaps. Write the sequence in a row and put the difference between each pair of neighbors underneath. If the differences follow a simple pattern, extend that pattern by one step and add it to the last term. If the differences don’t settle down, take the differences of the differences. Then, as a second test, look at the terms themselves and try to write each one as a product of two small numbers. A rule that you can see both ways is very likely the intended one.
Common Traps
A common slip is to notice that the numbers grow quickly and guess a doubling rule. It fits the jump from 6 to 12 and then falls apart. Another is to find the gaps correctly but extend the sequence with the last gap instead of the next one. A third trap is stopping at the first rule that seems to fit. With only five terms, many formulas can match, so a second, independent way of seeing the pattern is worth the extra minute.
Where This Puzzle Comes From
These are called pronic or oblong numbers, because each one can be drawn as a rectangle of dots that is one dot longer than it is wide. They are exactly twice the triangular numbers (1, 3, 6, 10, 15, 21), which the ancient Greeks studied as dot patterns. The method of taking differences is much older than any puzzle book, and it’s still a standard test: if the differences eventually become constant, the sequence follows a polynomial rule.
Try This Variation
Here is a cousin from the same family. What number comes next?
3, 8, 15, 24, 35, ?
Try both methods: the gaps, and writing each term as a product of two numbers.
Show the answer to the variation
48. The gaps are 5, 7, 9, 11, so the next gap is 13, and 35 + 13 = 48. As products: 1×3, 2×4, 3×5, 4×6, 5×7, so the next term is 6×8 = 48. Each term is also one less than a perfect square: 4, 9, 16, 25, 36, 49.