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30 Aug 2024, 08:36
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If the sum of all 21 terms of an 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. 11th smallest term is zero.
II. 11th largest term is zero.
III. The range of the numbers is 20.
A. I only
B. II only
C. III only
D. I and II only
E. None of the above
I. 11th smallest term is zero.
II. 11th largest term is zero.
III. The range of the numbers is 20.
A. I only
B. II only
C. III only
D. I and II only
E. None of the above
30 Aug 2024, 08:36
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Official Solution:
If the sum of all 21 terms of an 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. 11th smallest term is zero.
II. 11th largest term is zero.
III. The range of the numbers is 20.
A. I only
B. II only
C. III only
D. I and II only
E. None of the above
The sum of ALL 21 terms of an arithmetic progression being zero, implies that 10 terms are less than 0, 10 corresponding terms, with the same magnitude are greater than 0 and the middle term is 0.
The above would make the 11th smallest term equal to zero and the 11th largest term equal to zero. However, since we can stretch or shrink the sequence any way we'd like, the range—the positive difference between the largest and smallest terms—could be anything, not necessarily equal to 20.
Therefore, only options I and II must be true.
Answer: D
If the sum of all 21 terms of an 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. 11th smallest term is zero.
II. 11th largest term is zero.
III. The range of the numbers is 20.
A. I only
B. II only
C. III only
D. I and II only
E. None of the above
The sum of ALL 21 terms of an arithmetic progression being zero, implies that 10 terms are less than 0, 10 corresponding terms, with the same magnitude are greater than 0 and the middle term is 0.
The above would make the 11th smallest term equal to zero and the 11th largest term equal to zero. However, since we can stretch or shrink the sequence any way we'd like, the range—the positive difference between the largest and smallest terms—could be anything, not necessarily equal to 20.
Therefore, only options I and II must be true.
Answer: D
01 Oct 2024, 05:51
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Can we consider sequence of zeros as option to test? or sequence of zeros can't be AP?
Bunuel
