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 Square Sequence (Posted on 2003-12-19)
When you add all the terms up from this sequence: x² + (x-1)² + 3(x-2)² + (x-3)² + (x-4)² + (x-5)² + 3(x-6)² + (x-7)² ... it will be equal to half of (x³ + x² - x) for any positive even integer x. Prove why this works.

Example: 12² + 11² + 10² + 10² + 10² + 9² +8² + 7² + 6² + 6² + 6² + 5² + 4² + 3² + 2² + 2² + 2² + 1² if x = 12.

Note: The coefficients go 1, 1, then 3, then 3 1s, then 3, then 3 1s. The coefficients go in this order, even if there are coefficients left when the sequence stops. For example, with 6, the coefficients would go 1,1,3,1,1,1.

 See The Solution Submitted by Gamer Rating: 2.5000 (4 votes)

 Subject Author Date More about approach Gamer 2007-02-19 15:37:37 different approach Lee 2003-12-22 06:54:26 re: Sorry, my mistake Gamer 2003-12-20 20:41:47 Sorry, my mistake Penny 2003-12-20 10:39:36 re(3): proof Gamer 2003-12-20 07:36:11 re(2): proof Penny 2003-12-20 00:13:13 re: proof Gamer 2003-12-19 18:27:34 proof Charlie 2003-12-19 16:33:44 Solution (no computer assistance used) Penny 2003-12-19 15:33:16 if x=3??? rerun141 2003-12-19 12:38:22

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