[ Sequences (other).]

Can 2000 numbers be written in a row so that the sum of any three consecutive numbers is negative, while the sum of all the numbers is positive?

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Alternate 2 positive numbers with one negative.

## Solution

The number 2000 is not divisible by 3, and the overall sum can be positive (although the sum of any three consecutive numbers is negative) due to the first two and the last two numbers being sufficiently large positive numbers. Here is a corresponding example of a sequence of numbers: 1000, 1000, -2001, 1000, 1000, -2001, ..., 1000, 1000, -2001, 1000, 1000. In this sequence, there are 666 triplets (1000, 1000, -2001), and the last two numbers are 1000. Each triplet consists of two numbers equal to 1000 and one number equal to -2001, so the sum of any three consecutive numbers is -1. The sum of all numbers is \((-1) * 666 + 1000 + 1000 = 1334\) - a positive number.

## Answer

Yes, it is possible.