If the sum of first 51 terms of arithmetic progression is zero, then w

Question

If the sum of first 51 terms of arithmetic progression is zero, then which of the following must be true ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant)

I. 51st largest term is zero
II. 26th largest term is zero
III. All terms are non negative

A. I
B. II
C. III
D. I and II
E. None of the above

IanStewart · Answer

What is the source? The correct answer is not among the choices.

If you have an equally spaced list (an 'arithmetic progression'), then its average and median are equal. From the definition of the average, the sum of any list equals the number of terms times the average. So here, looking at only the first 51 terms, since the average equals the median ("m"), the sum is equal to 51m. So if the sum of the first 51 terms is 0, then 51m = 0 and the median m of the first 51 terms is 0. The median of the first 51 terms is the 26th term.

So we know the 26th term in this progression is zero, and since an arithmetic sequence either constantly goes up or constantly goes down (assuming the terms aren't all equal), then either the 26th smallest or 26th largest term is zero. But we have no way to tell which of those two cases we have, because we don't know if this sequence is increasing or decreasing, so there's no way to know if II is true. Item III can't be true unless every term is zero, which we don't know to be true, and item I could be true (the sequence has more than 51 terms in total, so it's possible that the 51st largest is zero) but will be false almost always, so the answer is 'none of the above'.

Bunuel · Answer

Thank you Ian. Edited the options. The source is unknown. Found the questions in the archives.

IanStewart · Answer

I'd guess the question writer meant to say that the sequence contains exactly 51 terms (in which case the 26th largest and 26th smallest terms are the same term) -- it's because the question says it's talking only about "the first 51 terms" of some longer progression that II is wrong.

Bunuel · Answer

Here are 3 new questions I wrote based on the above problem: https://gmatclub.com/forum/if-the-sum-o ... 98379.html https://gmatclub.com/forum/if-the-sum-o ... 98380.html https://gmatclub.com/forum/if-the-sum-o ... 98381.html

If the sum of first 51 terms of arithmetic progression is zero, then w

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If the sum of first 51 terms of arithmetic progression is zero, then w

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