If n is not 0 mod 3 we may easily find a sequence for which all sums of 3 successive terms are positive, but the sum of all the terms is negative.

First we take an obvious simplifying condition: The sequence must be periodic of period 3. Say the first 3 terms are a, b and c. We must have a b c > 0, so take a b c = 1.

Suppose now that n = 3k 1. Then we also require a k < 0. So take a = -k -1. Now we just need b c = k 2. So let, for example, b = k, c = 2.

On the other hand, if n = 3k 2, we need a b k < 0. So take a = -k, b = -1, c = k 2 (for example)
The method obviously generalises for similar problems, such as when the sum of every successive set of 4 terms must be positive.

Thankyou

sfwc
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