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 2 of 4: Exclude seventeen terms
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In plain words

Because 77 and 1111 are coprime, the two shifts weave every index into one contradiction cycle.

0<s11<s4<s15<s8<s1<s12<s5<s16<s9<s2<s13<s6<s17<s10<s3<s14<s7<00<s_{11}<s_4<s_{15}<s_8<s_1<s_{12}<s_5<s_{16}<s_9<s_2<s_{13}<s_6<s_{17}<s_{10}<s_3<s_{14}<s_7<0
Detailed analysis

If there were 1717 terms, the valid comparisons chain exactly as displayed: each index change is +11+11 or −7-7. It concludes 0<cdots<s7<00<cdots<s_7<0, impossible.