If the sum of the first 11 terms of an arithmetic progression

Question

If the sum of the first 11 terms of an arithmetic progression consisting 16 terms is zero, then which of the following COULD 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. 6th largest term is zero
III. All terms are non negative

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

gmatophobia · Accepted Answer

Observation- Its a  COULD  be true question and not MUST be true.

All the terms in the sequence can be zero.

Note: If all the terms in the sequence is same, the sequence is still in AP with common difference = 0

I. 11th smallest term is zero -  Yes 
II. 6th largest term is zero 0-  Yes 
III. All terms are non negative -  Yes

Option E

pudu · Answer

a=-5d then how it is possible to get all terms are positive? even if d =-1 then too we have negative terms...The answer is D...if not please correct me.

gmatophobia · Answer

pudu

I guess you're referring to statement 3.

Note, the statement doesn't state that the terms are positive. Rather, it states that - " All terms are  non negative  "

Hence, we now have the possibility of all terms = 0. So, if d = 0, all the terms can be zero in the series.

Hope that clarifies!

khumoyuns · Answer

5,4,3,2,1,0,-1,-2,-3,-4,-5,-6,-7,-8,-9,-10 - accounts for statements 1 and 2.

Also, all the terms in the sequence can be 0.

So answer is E

pudu · Answer

please someone helps me.if all the terms are zero then how it should be AP

gmatophobia · Answer

pudu

A set of numbers is in arithmetic progression if the difference between consecutive terms (common difference) is constant. That difference can be positive, negative or zero.

For example: 1,1,1,1 is in AP with common difference 0.

Posted from my mobile device

Kinshook · Answer

Asked: If the sum of the first 11 terms of an arithmetic progression consisting 16 terms is zero, then which of the following COULD be true ? (An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant)

Let the sequence be a-8d, a-7d, a-6d, ....a-d, a, a+d, .... a+7d.

The same of first 11 terms = a-8d + a-7d + a-6d + ...+a + a+d + a+2d = 11a - 33d = 0
a = 3d

I. 11th smallest term is zero : 
11th smallest term = a +2d = 3d + 2d = 5d
If d=0
COULD BE TRUE

II. 6th largest term is zero
6th largest term = a - 3d = 0
TRUE

III. All terms are non negative
a - 8d = 3d - 8d = - 5d
Which may be positive or negative
If a=10;d=-1
COULD BE TRUE

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

IMO E

vitaemaiores · Answer

Is d=0, a=0. Considered an AP?

Bunuel · Answer

Technically, yes.

Yes. If d = 0, then all terms are equal, so the difference between consecutive terms is constant. If a = 0 and d = 0, the sequence is:

0, 0, 0, ...

That is still an arithmetic progression.

Siddharth17 · Answer

Yes — an arithmetic sequence can have all elements equal to 0.

Here’s why:

An arithmetic sequence is defined by:

* First term a
* Common difference d

General form:
a, \; a + d, \; a + 2d, \; a + 3d, \dots

If you choose:

* a = 0
* d = 0

Then every term becomes:
0, \; 0, \; 0, \; 0, \dots

So:

* The sequence is constant
* The difference between consecutive terms is 0, which satisfies the definition of an arithmetic sequence

Conclusion:
Yes, a sequence of all zeros is a valid arithmetic sequence with common difference 0.

If the sum of the first 11 terms of an arithmetic progression

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If the sum of the first 11 terms of an arithmetic progression

Prep Toolkit

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

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