Let x1…xn be a finite sequence of real numbers such that: i) the sum of any 7 consecutive terms is always strictly positive; ii) the sum of any 13 consecutive elements is always strictly negative. Which of the following statements is true?
Pick one
Solution
Solution:
The answer is (A). It suffices to observe that if the sequence has more than 18 elements then the quantity: X=(x1+…+x7)+(x2+…+x8)+…+(x13+…+x19) is both strictly greater than zero and strictly less than zero; indeed: it is strictly greater than zero by condition (i), while it is less than zero since, upon rearranging the terms on the right-hand side, we have that: X=(x1+…+x13)+(x2+…+x14)+…+(x7+…+x19) which by (ii) is a quantity strictly less than zero. Hence the answer can only be (A).
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