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.
Many of the sequence problems can be solved by Induction. This problem is a proposition P(x) on the integers.
P(1)=2^2-1^2=3=1^2(2+1).
Assume P(k) for all x<=k.
Write P(k+1) in terms of P(k). There are two
more terms and one term has the sign reversed.
P(k+1)=P(k)+(2k+2)^2+(2k+1)^2-2(k+1)^2
=k^2(2k+1)+(2k+2)^2+(2k+1)^2-2(k+1)^2
=2k^3+7k^2+8k+3 after expansion and
collection.
Now compute P(k+1) by the formula.
P(k+1)=(k+1)^2(2(k+1)+1)
=(k+1)^2(2k+3)
=2k^3+7k^2+8k+3 after expansion and
collection. The two results are an identity.
Therefore the assumption implies P(k+1).
QED.
Proof by Induction
