# What is the sum of 2 2

### The sum of the first N square numbers

We consider the sum of the first N square numbers, i.e. 1 + 4 + 9 + ... + N^{2}.

total | Factorization | 3 * total | 2 * 3 * total | ||||||||||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 3 | |||

1+4 | 5 | 1 | 1 | 5 | 1 | 3 | 5 | 2 | 3 | 5 | |||

1+4+9 | 14 | 1 | 2 | 7 | 3 | 2 | 7 | 3 | 4 | 7 | |||

1+4+9+16 | 30 | 2 | 5 | 3 | 2 | 5 | 9 | etc | |||||

1+4+9+16+25 | 55 | 5 | 1 | 11 | 5 | 3 | 11 | ||||||

1+4+9+16+25+36 | 91 | 1 | 7 | 13 | 3 | 7 | 13 | ||||||

1+4+9+16+25+36+49 | 140 | 7 | 4 | 5 | 7 | 4 | 15 | ||||||

1+4+9+16+25+36+49+64 | 204 | 4 | 3 | 17 | 4 | 9 | 17 | ||||||

1+4+9+16+25+36+49+64+81 | 285 | 3 | 5 | 19 | 9 | 5 | 19 | ||||||

1+4+9+16+25+36+49+64+81+100 | 385 | 5 | 11 | 7 | 5 | 11 | 21 | ||||||

1+4+9+16+25+36+49+64+81+100+121 | 506 | 11 | 2 | 23 | 11 | 6 | 23 | ||||||

1+4+9+16+25+36+49+64+81+100+121+144 | 650 | 2 | 13 | 25 | 6 | 13 | 25 | ||||||

1+4+9+16+25+36+49+64+81+100+121+144+169 | 819 | 13 | 7 | 9 | 13 | 7 | 27 |

**Brahmagupta**(598-670),

in his book

*Brahmasphutasiddhanta*(628),

but without proof.

**Proofs:**

- 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 form 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:

- How does yoga help with pcos
- What use is Bhakthi Yoga
- How do I download my ustream video?
- Can I view statistics from other Quora users?
- Uses Memrise enough to learn German
- What attracts women
- Can a hardware engineer become a software engineer
- When will our life on earth end?
- What is the root of character flaws
- Why does Facebook not have a BNBR policy
- Why don't blues singers sing about Prozac
- Why do some songs have instrument solos?
- What are FX Market Systems
- How do I do Yolo on Snapchat
- Is Ben Shapiro too prejudiced against Israel?
- How could I solve this
- Can I make ceviche with frozen fish
- Ant Man is the most powerful avenger
- Are 5g routers safe
- Who was Homer
- Will the Conservatives ever reject Donald Trump's leadership?
- Which is better JavaFX or Swing
- What are examples of top speed
- How should I download watchfree su videos