M41-57

Question

If the sum of the first 31 terms of an arithmetic progression consisting of 46 terms is zero, then which of the following MUST be true? (An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant.)
 
I. The 31st smallest term is zero.
 
II. The 16th largest term is zero.
 
III. The sum of the largest and smallest terms of the sequence is positive.

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

Bunuel · Answer

Official Solution:

If the sum of the first 31 terms of an arithmetic progression consisting of 46 terms is zero, then which of the following MUST be true? (An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant.)
 
I. The 31st smallest term is zero.
 
II. The 16th largest term is zero.
 
III. The sum of the largest and smallest terms of the sequence is positive.

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

Firstly, note that we don't know whether the sequence is increasing or decreasing.
 
If the sequence is increasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:
   
 a_1 \leq ... \leq (a_{16} = 0) \leq ... \leq a_{31} \leq ... \leq a_{46} 
   
If the sequence is decreasing, then the fact that the sum of the first 31 terms of the arithmetic progression is zero would imply that:
   
 a_1 \geq ... \geq (a_{16} = 0) \geq ... \geq a_{31} \geq ... \geq a_{46} 
   
As we can see, none of the options MUST be true.

Answer: E

Rebaz · Answer

Bunuel

If i apply the same logic in https://gmatclub.com/forum/m41-435049.html#p3447659, then I and II must be true.

I wonder why I and II are not true, coz the question is almost the same?

Thanks in advance!

Bunuel · Answer

In the  other question  we have:

If the sum of  ALL  21 terms of an arithmetic progression is zero/

In  this question  we have:

If the sum of the  FIRST  31 terms  of an arithmetic progression consisting of 46 terms is zero.

phmahi1997 · Answer

Correct me if I am wrong:

For decreasing sequence, the largest 16th value would be zero!
But for Increasing sequence, the smallest 16th value would be zero but largest 16th value should be any other value than zero? am I right?

Bunuel · Answer

Yes, you are correct.

vestenhaus · Answer

This is a great question that’s helpful for learning. didn't consider that arithmetic progression can be decreasing

Unknown5612 · Answer

AP can also be 0,0,0...(all terms zero), right?

Bunuel · Answer

Yes, that's one of the possible cases.

GG27 · Answer

Hey @Bunnuel could you please explain how III isn't always true?

Bunuel · Answer

Sure. The simplest example is a sequence consisting entirely of 0's, its sum is 0, which is not positive. Another example: consider the sequence 15, 14, 13, ..., (a16 = 0), -1, -2, ..., -30. Here, the largest element is 15, the smallest is -30, and their sum is 15 + (-30) = -15, which is negative.

Gg153 · Answer

Bunuel Bunuel I apologize if this is a stupid question, but shouldn't the 31st smallest term and 16th largest term be the same? if we have a,b,c,d,e such that ad>c>b>a) and in the case of the question both must be zero Please correct me

Bunuel · Answer

Consider the following sequences.

Decreasing sequence: 
15, 14, 13, ..., (a16 = 0), -1, -2, ..., -30
• The 31st smallest term is 0.
• The 16th largest term is 0.
• The sum of the largest and smallest terms of the sequence is negative.

Increasing sequence: 
-15, -14, -13, ..., (a16 = 0), 1, 2, ..., 30
• The 31st smallest term is 15.
• The 16th largest term is 15.
• The sum of the largest and smallest terms of the sequence is positive.

ritzysharma · Answer

Just an alternate solution, as I was not able to visualize this while attempting.

If we dont assume any increasing or decreasing sequence as you have done and take first term as a, common difference as d. 
We are given that first 31 terms are 0, using sum of AP formula, it would mean "31*(a+15d)=0" which translates to a=-15d.

1. means a31=a+30d= -15d+30d = 15d, now d can be +/-/0 any of these are possible. 
2. 16th largest term is also a31 (same as above)
3. Sum of largest and smallest terms = a+45d+a = 2a+45d = -30d+45d = 15d (again same as 1 and 2)
Hence, none of 1,2 and 3 need to be must be true.

Bunuel  is this okay or any flaw in the approach?

Bunuel · Answer

Your logic is not entirely correct.

In your solution, you labeled a31 as “the 31st term,” but the problem statement uses “31st smallest” and “16th largest”. That depends on whether the sequence is increasing or decreasing:

If the sequence is increasing: the 31st smallest is indeed a31, and the 16th largest is also a31 (since the largest is a46, 16th largest = a31). 
 If the sequence is decreasing: the 31st smallest is a16 (not a31), and the 16th largest is a16 (since largest is a1, 16th largest = a16).

Please review the discussion above for more.

M41-57

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