What number should come next? 10, 11, 15, 24, 40, ?

Question

What number should come next? 10, 11, 15, 24, 40, ?

1. A. 65 (Correct answer)
2. B. 60
3. C. 55
4. D. 50

Correct Answer

Option A — 65

Detailed Solution & Explanation

The correct answer is 65.

Key Points

• This is a difference-of-squares series — the gaps between terms are consecutive perfect squares.
• Series: 10, 11, 15, 24, 40, ?
• 11 − 10 = 1 = 1²
• 15 − 11 = 4 = 2²
• 24 − 15 = 9 = 3²
• 40 − 24 = 16 = 4²
• The next difference is 5² = 25, so the term is 40 + 25 = 65.
• compute first differences before anything else.
• Here they are 1, 4, 9, 16, 25 — the perfect squares in order — a very common hidden structure.

Exam Tip

• distinguish this from q49-type series whose differences are odd numbers (3, 5, 7, 9…); the quickest diagnostic is to write the differences in a row and see whether they are squares, odds, or a multiplied pattern.

Additional Information

• The general term follows from the structure: each step adds the next perfect square, so the *n*th term is 10 + (1² + 2² + … + (n−1)²). Using the standard sum of squares k(k+1)(2k+1)/6, the 6th term is 10 + (5·6·11)/6 = 10 + 55 = 65.
• Verify by extending: the next difference would be 6² = 36, giving 65 + 36 = 101, and then 7² = 49 → 150. A rule that keeps producing whole terms consistently is almost certainly the intended one.
• The three difference patterns that cover most exam series are worth separating: constant differences (arithmetic), perfect squares 1, 4, 9, 16 as here, and odd numbers 3, 5, 7, 9 — the last of which generates the squares themselves (1, 4, 9, 16, 25).
• The reliable method is mechanical: write the first differences in a row under the series. If they are not obviously patterned, take the second differences. Here the second differences are 3, 5, 7, 9 — consecutive odd numbers — which confirms the squares reading.
• Watch the distractors: 60 and 55 come from adding 20 or 15 (misreading the gap as arithmetic), and 50 from adding 10. All three assume a constant or near-constant difference, which the first four gaps already rule out.