Square Numbers

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Numbers of the form n², produced by multiplying an integer by itself.

A square number (or perfect square) has the form S_n = n² where n is a non-negative integer. The sequence begins 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Consecutive square numbers differ by consecutive odd numbers: (n+1)² − n² = 2n + 1. This gives a simple way to recognize the pattern: if the differences between terms form the sequence 3, 5, 7, 9, ..., you are looking at squares.

S_n = n²; S_{n+1} − S_n = 2n + 1

• 1, 4, 9, 16, 25 → next is 36 (6²)
• 0, 1, 4, 9, 16 → next is 25 (5²)
• 4, 9, 16, 25, 36 → differences 5, 7, 9, 11 show odd-number gaps

How to recognize it

• Check whether each term is a perfect square of an integer
• Compute first differences; if they form 3, 5, 7, 9, ... you have squares
• Second differences are constant at 2

Common mistakes

• Confusing n² with 2n (doubling rather than squaring)
• Forgetting that 0 and 1 are also valid square numbers
• Mistaking squares for cubes when small values overlap (e.g. 1, 8 both appear)

Practice guidance

Number Sequences