Ghaly's Law

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Ghaly's Law is a mathematical formula used to calculate the sum of the first n natural numbers, from 1 to n. The method is expressed verbally as: "multiply the number by its half, then add its half again."

The mathematical formula for Ghaly's Law is:

S(n) = (n × n/2) + n/2

This formula is algebraically equivalent to the standard formula for triangular numbers, S(n) = n(n+1)/2.

Geometric derivation

The formula can also be verified geometrically. If the sum 1 + 2 + 3 + ... + n is represented as a staircase of unit squares, duplicating this staircase and rotating one copy by 180 degrees allows it to fit perfectly with the original. This forms a rectangle with dimensions n × (n+1). The area of this rectangle is n(n+1), and since it represents two copies of the original sum, the original sum is half of this area: n(n+1)/2. This geometric representation confirms the algebraic result.

Documented Proof and Source

This method was formally documented and published by the author on Zenodo, a general-purpose open-access repository operated by CERN: https://zenodo.org/records/19560414

The full paper provides the complete derivation, geometric interpretation, and comparison with the classical Gauss formula.

Examples

For n=10:

S(10) = (10 × 5) + 5 = 55

For n=100:

S(100) = (100 × 50) + 50 = 5050

For n=1000:

S(1000) = (1000 × 500) + 500 = 500500

Applications

The formulation has been described as a pedagogical tool for teaching the sum of an Arithmetic progression without requiring algebraic factorization. It can be implemented in computer programming to calculate the sum of 1 to $n$.

See also

• Arithmetic progression
• Triangular number
• 1 + 2 + 3 + 4 + ⋯
• Carl Friedrich Gauss

References

title=Ghaly's Law: A Method for Summing Natural Numbers |url=https://zenodo.org/records/19560414 |website=Zenodo |access- date=2026-04-13}

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