MathLabs

Problem 2

In a finite sequence of real numbers, every sum of seven successive terms is negative and every sum of eleven successive terms is positive. Determine the maximum length.
Step 1 of 4: Introduce partial sums
Previous stepNext step
In plain words

Sliding windows become comparisons between endpoints of partial sums.

s0=0,quadsj=x1+cdots+xjs_0=0,quad s_j=x_1+cdots+x_j
Detailed analysis

For every admissible index, a negative seven-term sum gives sj+7<sjs_{j+7}<s_j, while a positive eleven-term sum gives sj+11>sjs_{j+11}>s_j.