In a certain sequence, the next term is found by taking the number before it minus the number two numbers before it.
For example, in the sequence a, b, c, d... c = b-a, d = c-b, and so on.
Starting with 54 and 93, what would be the sum of the first six thousand terms?
Let the first two terms be x and y.
Then, in conformity with the tenets inclusive of the problem under reference:
Third Term is y-x; Fourth term is -x; Fifth term is -y while the Sixth Term is x-y.
So, summing over every sixth term we would always obtain 0 as a result.
Accordingly, it can be said that in general:
(i) The sum of the first 6N terms is 0.
(ii) The sum of the (6P+1) th through 6Qth term is 0, whenever Q>P and both P and Q are positive integers.
It is thus a trivial matter to substitute N = 1000 in (i) and obtain zero as the required solution to the given problem.
Puzzle Solution: An Attempt to generalisation
