If you subtract the sum of the first x numbers, squared, from the sum of the next x numbers, squared, (for example, 6² + 5² + 4² - 3² - 2² - 1²); you will get (x²)(2x+1) and also x²(x+1)² minus x^4. Prove why this works.

both solutions are correct (Zaphod and Larry).
Now I give the 3-rd one which is based on formula of the sum of the first n squared numbers:
1² + 2² +....+ n² = n(n+1)(2n+1)/6
The sequence may be expressed as:
(1² + 2² + ... + x² + (x+1)² + ... + 2x²) -
2(1² + 2² + ... + x²) =
2x(2x+1)(4x+1)/6 - 2x(x+1)(2x+1)/6 =
x²(2x+1)
QED
The last one is elementary:
x²(2x+1) = x²(x+1)-x^4.

Posted by luminita on 2003-12-28 14:56:43