Triangular Numbers

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Sums 1 + 2 + 3 + ... + n that count objects arranged in a triangle.

The nth triangular number is T_n = n(n+1)/2, the sum of the first n positive integers. The sequence begins 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66. Each term adds the next integer to the previous term, so first differences grow as 2, 3, 4, 5, ... and second differences are constant at 1. Triangular numbers show up when counting handshakes, grid diagonals, and round-robin pairings.

T_n = n(n+1)/2

• 1, 3, 6, 10, 15 → next is 21 (6 + 15)
• 3, 6, 10, 15, 21 → next is 28 (add 7)
• 6, 10, 15, 21, 28 → differences 4, 5, 6, 7 (consecutive integers)

How to recognize it

• Compute first differences; if they form 2, 3, 4, 5, 6, ... you have triangular numbers
• Second differences are constant at 1
• Check if each term matches n(n+1)/2 for some integer n

Common mistakes

• Confusing with square numbers (which have first differences 3, 5, 7, 9, ...)
• Forgetting the division by 2 and writing n(n+1) instead of n(n+1)/2
• Missing the off-by-one: T_1 = 1, not 0

Practice guidance

Number Sequences