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 3 of 4: Construct sixteen terms
Previous stepNext step
In plain words

An arbitrary set of partial sums with the required order automatically yields the desired sequence by differences.

s10<s3<s14<s7<0<s11<s4<s15<s8<s1<s12<s5<s16<s9<s2<s13<s6s_{10}<s_3<s_{14}<s_7<0<s_{11}<s_4<s_{15}<s_8<s_1<s_{12}<s_5<s_{16}<s_9<s_2<s_{13}<s_6
Detailed analysis

Choose real numbers satisfying this strict ordering, and define x1=s1x_1=s_1 and xj=sj−sj−1x_j=s_j-s_{j-1} for jge2jge2.