7.4. A row contains 100 numbers. Is it necessarily true that the sum of all these numbers is positive if it is known that a) the sum of any seven numbers is positive? b) the sum of any seven consecutive numbers is positive?
Answer: a) necessarily; b) no. Solution. a) Let $a_{1}, a_{2} \ldots a_{100}$ be the given numbers. We compute 100 positive sums $S_{1}=a_{1}+a_{2}+\ldots+a_{7}, S_{2}=a_{2}+a_{3}+\ldots+a_{8}, \ldots, S_{100}=a_{100}+a_{1}+\ldots+a_{6}$ (seven consecutive numbers are added in a cyclic arrangement). Then $S_{1}+S_{2}+\ldots+S_{100}=7 S$, where $S$ is the sum of all numbers (since each number is counted in seven sevens). Therefore, $S>0$.

b) Take such numbers $a_{1}=a_{8}=a_{15}=\ldots=a_{92}=a_{99}=-6$ (every seventh number is -6) and let the other numbers be 1.01. Then in any seven consecutive numbers, the sum is $-6+1.01 \cdot 6=0.06>0$, and the sum of all 100 numbers is $(-6) \cdot 15+(1.01) \cdot 85=-4.15<0$.