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 2 of 5: Sum the array by rows
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S=∑i=17∑j=010ai+j<0(by rows)S=\sum_{i=1}^{7}\sum_{j=0}^{10}a_{i+j}<0\quad\text{(by rows)}
Detailed analysis

Every row is the sum of eleven consecutive terms, so the hypothesis makes every row sum negative. Therefore the total sum SS of all entries is negative.