If the sum of all 21 terms of an arithmetic progression is zero, then

Question

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.

I. 11th smallest term is zero
II. 11th largest term is zero
III. The range of the numbers is 20

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

Bunuel · Accepted Answer

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.

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

chetan2u · Answer

In an AP, the median = mean

So, the median of these 21 terms is the 11th term and is equal to mean. 
Mean is 0, as the sum is 0.

I. 11th smallest term is zero
The middle term is 11th, so it will be the same from top and bottom. So 0 is the correct answer.

II. 11th largest term is zero
Same as above.

III. The range of the numbers is 20
Range could be anything. 
The series could be -10, -9, -8,…..0…. 8, 9, 10 or -20, -18, -16………0……., 16, 18, 20.

I and II are correct.

D

KarishmaB · Answer

In an AP of 21 terms, average of AP will be the middle term i.e. 11th term.

Sum of an AP = n * Average of AP

Since Sum = 0, and n = 21, Average of AP = 0.
So 11th term must be 0 whether we arrange the AP in increasing or decreasing order.

Answer (D)

reyrest · Answer

Can all the terms in AP be 0? with a constant of 0? If so, isn't it option E?

Bunuel · Answer

If all terms are 0, aren't II and III true?

reyrest · Answer

If all terms are 0, can we say if one is larger or smaller than the other? Because its a MUST true question and not could be true question.

somewhat similar question below but its a could be question and not MUST BE.
https://gmatclub.com/forum/if-the-sum-o ... 8379.html#

Bunuel

If all terms are 0, aren't II and III true?

Bunuel · Answer

________________________________
Yes.

reyrest · Answer

I'm sorry, I'm still not convinced on how we can say one term is larger or smaller than the other term if all the terms in a sequence are equal. Could you please help me understand?

Bunuel · Answer

If all terms are equal, then every term is both smallest and largest at the same time. So the 11th smallest and 11th largest just refer to the 11th term in order, which is still 0.

If the sum of all 21 terms of an arithmetic progression is zero, then

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If the sum of all 21 terms of an arithmetic progression is zero, then

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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