harsh.champ wrote:shashank.ism wrote:If a1, a2, a3 are the first three terms of an arithmetic progression containing 100 terms, what is the possible common ratio when a2, a1 and a3 are in geometric progression?
a) - 2
b) 2
c) 1
d) both (1) and (3)
e) impossible
First of all,I would like to point that 2 more alternatives of this question can appear.
This question can be rephrased as :-
If a1, a2, a3 are the first three terms of an arithmetic progression containing 100 terms, what is the possible
common difference of the A.P. when a2, a1 and a3 are in geometric progression?
Or
If a1, a2, a3 are the first three terms of an
geometric progression containing 100 terms, what is the possible common ratio when a2, a1 and a3 are in
arithmetic progression?
Now,solving the initial question:-
So,let the the no.s be (a2 - n),a2,(a2 + n).
Since they are in G.P.,hence , (a2)^2 = (a2 - n)(a2 + n )
hence, n=0.
So,all the numbers are same.
Hence,the
common ratio is 1.Hence,C
The (D) answer choice should be
both (A) and (C) not
both (1) and (3).
The numeric option choice can be confusing at times.
First, none of the above questions could appear as written on the GMAT; you don't need to know what the phrase 'geometric progression' means for the test. If you see a geometric sequence on the test, they'll explain what it is using different language. The wording is also somewhat backwards in the question; it should state as a fact that a_2, a_1, a_3 form a geometric progression, and then ask what the common ratio could be if a_1, a_2, a_3 are in arithmetic progression. It also needs to make clear that the terms are nonzero; otherwise the ratio can be anything at all.
Second, there is a mistake when you write: (a2)^2 = (a2 - n)(a2 + n ). I'm not sure how you arrived at this; I can only guess that you've assumed that the first term in the progression will be equal to the product of the second and third terms. That's not normally true, as you can see by looking at any familiar geometric series.
Finally, it should be clear that the terms can be equal, so '1' is a possible answer here. Now, is it possible that -2 is the ratio? We could just take a numerical example; a_2 is the first term of our geometric sequence, so we could make it equal to 1, making a_1 = -2 and a_3 = 4. We can verify that a_1, a_2, a_3 now forms an arithmetic sequence: -2, 1, 4. So the ratio could be -2.
Or we can prove this more formally: we know a_2, a_1, a_3 form a geometric sequence, so in this case we would have
a_1 = (-2)a_2
a_3 = (-2)a_1 = 4a_2
Now, a_1, a_2, a_3 needs to be an arithmetic sequence; that is, (-2)a_2, a_2, 4a_2 must form an arithmetic sequence. It clearly does; the difference between consecutive terms is 3a_2.
So D is the correct answer (we don't need to bother checking '2' since there is no answer choice 'all of the above').
