Combinatorics question See attached image A computer program

Combinatorics question. See attached image

A computer programmer claims that he generated six real numbers a_1, a_2, . . . , a_6 so that the sum of any four consecutive a_i is positive, but the sum of any three consecutive a_i is negative. Prove that his claim is false.

Solution

The six real numbers which the programmer has generated be a₁,a₂,a₃,a₄,a₅,a₆.
According to the question, let a₁+a₂+a₃+a₄ = k₁ ... Eq (1) [where k₁ is a positive real number]
Let a₁+a₂+a₃ = -p₁ ... Eq (2) [where p₁ is a positive real number].
Subtracting Eq (2) from Eq (1), we get:
a₄ = k₁ + p₁ (which is positive as k₁ and p₁ both are positive)

Similarly let a₂+a₃+a₄+a₅ = k₂ ... Eq (3) [where k₂ is a positive real number]
Let a₂+a₃+a₄ = -p₂ ... Eq (4) [where p₂ is a positive real number]
Subtracting Eq (4) from Eq (3), we get:
a₅ = k₂ + p₂ (which is positive as k₂ and p₂ both are positive)

Similarly we will get the following results:
a₆ = k₃ + p₃
a₁ = k₄ + p₄
a₂ = k₅ + p₅
a₃ = k₆ + p₆

Now all the numbers from a₁ to a₆ are coming out to be positive from the given data. So the sum of any three numbers cannot be negative. Thus his claim is false.