385 Is a Square Pyramidal Number

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:

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