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Pattern Recognition: What comes next in the sequence 2, 6, 12, 20, 30?

Problem Statement

Look at the following sequence of numbers: 2, 6, 12, 20, 30, ? What number comes next?

Answer

42

Step-by-Step Explanation

  1. 1

    Let's examine the differences between consecutive numbers: 6-2=4, 12-6=6, 20-12=8, 30-20=10

  2. 2

    The differences form a pattern: 4, 6, 8, 10. These are consecutive even numbers.

  3. 3

    Following this pattern, the next difference should be 12.

  4. 4

    Therefore, the next number in the sequence is 30 + 12 = 42.

  5. 5

    We can verify this pattern by looking at the formula: each number is n(n+1) where n starts at 1.

  6. 6

    Position 1: 1×2 = 2, Position 2: 2×3 = 6, Position 3: 3×4 = 12, Position 4: 4×5 = 20, Position 5: 5×6 = 30, Position 6: 6×7 = 42

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