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0|0|0|||1||Signs and sums in a sequence|sfwc||14:55:05|05/30/2007|A note on http%3A%2F%2Fwww.cut-the-knot.org%2FCurriculum%2FAlgebra%2FNegPosSum.shtml%0D%0A%0D%0AIf n is not 0 mod 3 we may easily find a sequence for which all sums of 3 successive terms are positive%2C but the sum of all the terms is negative. %0D%0A%0D%0AFirst we take an obvious simplifying condition%3A The sequence must be periodic of period 3. Say the first 3 terms are a%2C b and c. We must have a   b   c %3E 0%2C so take a   b   c %3D 1. %0D%0A%0D%0ASuppose now that n %3D 3k   1. Then we also require a   k %3C 0. So take a %3D -k -1. Now we just need b   c %3D k   2. So let%2C for example%2C b %3D k%2C c %3D 2. %0D%0A%0D%0AOn the other hand%2C if n %3D 3k   2%2C we need a   b   k %3C 0. So take a %3D -k%2C b %3D -1%2C c %3D k   2 %28for example%29%0D%0AThe method obviously generalises for similar problems%2C such as when the sum of every successive set of 4 terms must be positive.%0D%0A%0D%0AThankyou%0D%0A%0D%0Asfwc%0D%0A%3C%3E%3C
1|1|0|||||RE%3A Signs and sums in a sequence|alexb||09:08:40|05/31/2007|Yes%2C the problem is pretty simple%2C although at first site it may sound quite improbable to an untrained mind.