MathLabs

Problem 5

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.
Step 4 of 5: Choose a 16-term periodic-pattern candidate
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(−a,−a,b,−a,−a,−a,b,−a,−a,b,−a,−a,−a,b,−a,−a),−5a+2b>0,−8a+3b<0(-a,-a,b,-a,-a,-a,b,-a,-a,b,-a,-a,-a,b,-a,-a),\qquad -5a+2b>0,\quad -8a+3b<0
Detailed analysis

Use the displayed 16-term pattern, a truncation of a cycle of seven entries. Every seven consecutive terms have sum −5a+2b-5a+2b, and every eleven consecutive terms have sum −8a+3b-8a+3b. Thus it is enough to choose nonzero integers satisfying −5a+2b>0-5a+2b>0 and −8a+3b<0-8a+3b<0.