## Signs and Sums in a Sequence

In a sequence of numbers, the sum of any successive three terms is positive. Is it possible for the total of all the terms to be negative? [Shapovalov, p. 15].

The applet allows you to experiment with the problem. The numbers in the sequence (the bottom row) are clickable. They may be increased or decreased by clicking off their central line.

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### References

- A. Shapovalov, The Principle of Narrow Passages (in Russian), ISBN 5-94057-234-0, 2006, pp 24.

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In a sequence of numbers, the sum of any successive three terms is positive. Is it possible for the total of all the terms to be negative?

### Solution

The answer (yes or no) depends on the length (say N) of the sequence. If N is divisible by 3, then the answer is an unqualified No. Indeed, in this case, the total consists of the sum of the terms taken by three.

Let's consider the case where N = 1 (mod 3). We may split the sequence into successive triples in a variety of ways each, time leaving a single term which of necessity must be negative. This makes the terms in positions 1, 4, 7, ..., N negative. The rest may be positive. Assume all the negative terms equal -x,

-x + 2y > 0,

-Kx + (N - K)y < 0,

Where K = [N/3] + 1, the brackets being a symbol for the floor function. Note that

(1) |
2y > x and x < [(N - K)/K]y |

the problem will be solved. But there's a plenty of pairs

If N = 2 (mod 3) the consideration does not flow that smoothly. However, solutions do appear to exist. E.g., for

The problem is actually simpler than it sounds. W. McWorter observed that the sequence

Nathan Bowler has also remarked that 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

Suppose now that

On the other hand, if

### Remark

In a circular arrangement of numbers, the situation is entirely different. If we add the sums of all the successive triples, the result will be three times the total, so that the latter is necessarily positive if all such sums are.

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