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Math Expert V
Joined: 02 Sep 2009
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An arithmetic sequence is a sequence in which each term after the firs  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 47% (03:06) correct 53% (02:34) wrong based on 32 sessions

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An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant, which is also known as the common difference of that arithmetic sequence. The sequence S contains all the terms of two different increasing arithmetic sequences P and Q such that the number of terms in sequence S is equal to the sum of the number of terms in sequences P and Q. If each of the arithmetic sequences P and Q consists of 10 positive integral terms, how many distinct terms does sequence S have?

(1) The least common multiple of the common differences of the sequences P and Q is 6

(2) The third term of the sequence P is equal to the second term of the sequence Q

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An arithmetic sequence is a sequence in which each term after the firs  [#permalink]

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Bunuel wrote:
An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant, which is also known as the common difference of that arithmetic sequence. The sequence S contains all the terms of two different increasing arithmetic sequences P and Q such that the number of terms in sequence S is equal to the sum of the number of terms in sequences P and Q. If each of the arithmetic sequences P and Q consists of 10 positive integral terms, how many distinct terms does sequence S have?

(1) The least common multiple of the common differences of the sequences P and Q is 6

(2) The third term of the sequence P is equal to the second term of the sequence Q

Question: Number of distinct terms in S = ?

Statement 1: LCM of common differnece = 6

The common differences may be {1, 6} or {2, 3} or {2,6} or {3, 6}

So common terms count may differ hence

NOT SUFFICIENT

Statement 2: The third term of the sequence P is equal to the second term of the sequence Q

But afterteh common term, there may not be any common terms or there may be some common terms hence

NOT SUFFICIENT

COmbinig the statements

Case 1:(Common difference in sequence P = 2 and in Q = 6)
Terms in P may be {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
Terms in Q may be {0, 6, 12, 18, 24, 30, 36, 42, 48, 54}
i.e. 2 common terms {6, 12, 18}

Case 2: (Common difference in sequence P = 2 and in Q = 3)
Terms in P may be {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
Terms in Q may be {3, 6, 9, 12, 15, 18, 21, 24, 27, 30}
i.e. 3 common terms {6, 12, 18}

Case 3: (Common difference in both sequence = 6)
Terms in P may be {-6, 0, 6, 12, 18, 24, 30, 36, 42, 48}
Terms in Q may be {0, 6, 12, 18, 24, 30, 36, 42, 48, 54}
i.e. 9 common terms i.e. only 11 distinct terms

Number of common terms is varying i.e Number of Distinct terms is also varying hence

NOT SUFFICIENT

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Originally posted by GMATinsight on 25 May 2020, 04:28.
Last edited by GMATinsight on 25 May 2020, 06:03, edited 1 time in total.
GMAT Tutor P
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An arithmetic sequence is a sequence in which each term after the firs  [#permalink]

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Bunuel wrote:
An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant, which is also known as the common difference of that arithmetic sequence. The sequence S contains all the terms of two different increasing arithmetic sequences P and Q such that the number of terms in sequence S is equal to the sum of the number of terms in sequences P and Q. If each of the arithmetic sequences P and Q consists of 10 positive integral terms, how many distinct terms does sequence S have?

(1) The least common multiple of the common differences of the sequences P and Q is 6

(2) The third term of the sequence P is equal to the second term of the sequence Q

Using both statements, the common differences in P and Q can both be 6. Then using Statement 2, our sequences could be something like:

P: 10, 16, 22, 28, ....
Q: 16, 22, 28, 34, ...

These overlap almost completely; only the first term of P is not in Q, and only the last term of Q is not in P. So we'll only have 11 distinct values (the nine overlapping ones, plus the two unique values) if we combine P and Q.

But the common difference might be 1 in sequence P, and 6 in sequence Q. Then we could have

P: 20, 21, 22, 23, 24, ...
Q: 16, 22, 28, 34, ...

which clearly do not overlap nearly as much as in the first example, so we'll have more than 11 distinct values here and the answer must be E.

There are several issues with the question, though. For one, it's much wordier than real GMAT questions, and for another, it asks us to consider the Least Common Multiple of numbers that can be negative (common differences need not be positive). I'd expect in this kind of question for sequence S to be well-defined, but it's not, since we don't know how it is ordered, and there are some other issues with the wording. So I don't think any test taker who finds this question confusing should be concerned about that.
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Re: An arithmetic sequence is a sequence in which each term after the firs  [#permalink]

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Bunuel wrote:
An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant, which is also known as the common difference of that arithmetic sequence. The sequence S contains all the terms of two different increasing arithmetic sequences P and Q such that the number of terms in sequence S is equal to the sum of the number of terms in sequences P and Q. If each of the arithmetic sequences P and Q consists of 10 positive integral terms, how many distinct terms does sequence S have?

(1) The least common multiple of the common differences of the sequences P and Q is 6

(2) The third term of the sequence P is equal to the second term of the sequence Q

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S includes sequence P and Q but we know nothing else about S. We would like to know if P and Q have any overlaps. If no overlaps, then S must have 20 distinct terms, 10 from P and 10 from Q. If there is any overlap between P and Q, then P and Q combined would be 19 distinct terms or less, and for S we wouldn't know if the 20th term would be a unique term or not.
Thus the only way for sufficiency is to prove P and Q have no overlap.

Statement 1:
The common differences have a relation, but P and Q can still possibly have no overlaps. We could also have some overlapping terms so insufficient.

Statement 2:
We can confirm P and Q must have at least one overlapping term. However we still can't answer how many distinct terms S has, we know it doesn't have to be 20.

Combined:
Same idea from statement 2. Insufficient.
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Re: An arithmetic sequence is a sequence in which each term after the firs  [#permalink]

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# An arithmetic sequence is a sequence in which each term after the firs  