What is the sum of 1 3
The sum of the first N square numbersWe consider the sum of the first N square numbers, i.e. 1 + 4 + 9 + ... + N2.
|total||Factorization||3 * total||2 * 3 * total|
in his book Brahmasphutasiddhanta (628),
but without proof.
- Algebraic: With full induction
- Geometric proof (by Giorgio Goldoni):
Build 6 pyramids of the following shape (here for N = 4):
They can be put together to a cuboid with the edge lengths N, N + 1, 2N + 1.
Here is the assembly of three such pyramids:
A cuboid "with an outside staircase" is obtained.
Obviously, two such cuboids with their external stairs together form a compact cuboid!
For capital N, these pyramids resemble those pyramids known from dividing the cube into thirds by congruent pyramids:
In Chinese these are called pyramids Yang-ma, they play an important role, for example, in calculating the volume of truncated pyramids (Liu Hui, 3rd century CE, commentary on the 9 chapters).
The above pyramids, which we used in proving the formula for the sum of the first N square numbers, generalize the geometric proof for the sum of the first N numbers.
Here the case N = 5:
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