385 is the sum of the squares of the first ten counting numbers. Let me demonstrate what that means. All of the following are square pyramidal numbers:

- 1² = 1
- 1² + 2² = 5
- 1² + 2² + 3² = 14
- 1² + 2² + 3² + 4² = 30
- 1² + 2² + 3² + 4² + 5² = 55
- 1² + 2² + 3² + 4² + 5² + 6² = 91
- 1² + 2² + 3² + 4² + 5² + 6² + 7² = 140
- 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² = 204
- 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² = 285
- 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² = 385

We could have found the total, 385, much more quickly by putting the number ten in for “n” in this formula

Which means the sum of the first ten square counting numbers = (10 x 11 x 21)/6 = 5 x 11 x 7 = 385. (Notice its prime factorization!)

Here’s a video explanation:

Thanks to solvemymaths for bringing the video to my attention. It clearly explains why the formula works.

Here is the factoring information for 385:

- 385 is a composite number.
- Prime factorization: 385 = 5 x 7 x 11
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 385 has exactly 8 factors.
- Factors of 385: 1, 5, 7, 11, 35, 55, 77, 385
- Factor pairs: 385 = 1 x 385, 5 x 77, 7 x 55, or 11 x 35
- 385 has no square factors that allow its square root to be simplified. √385 ≈ 19.621

This is so cool.

I thought so, too!

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