a1, a2, a3, ... an is a sequence of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative. What is the largest possible value for n?
Solution
Answer: 16.
We cannot have 17 terms, because then:
a1 + a2 + ... + a11 < 0 a2 + a3 + ... + a12 < 0 a3 + a4 + ... + a13 < 0 ... a7 + a8 + ... + a17 < 0So if we add the inequalities we get that an expression is negative. But notice that each column is positive. Contradiction.
On the other hand, a valid sequence of 16 terms is: -5, -5, 13, -5, -5, -5, 13, -5, -5, 13, -5, -5, -5, 13, -5, -5. Any run of 7 terms has two 13s and five -5s, so sums to 1. Any run of 11 terms has three 13s and eight -5s, so sums to -1.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002