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

Question

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. An increasing arithmetic sequence N consists of a set of distinct negative integers and an increasing arithmetic sequence P consists of a set of distinct positive integers. The sequence C contains all the terms of arithmetic sequences N and P such that the number of terms in sequence C is equal to the number of terms in arithmetic sequences N and P. Is sequence C an arithmetic sequence?

(1) The sum of the largest term of the sequence N and the smallest term of the sequence P is zero.

(2) For every integer in sequence N, there exists an integer in sequence P with the same magnitude.

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Official Answer

E

(1) The sum of the largest term of the sequence N and the smallest term of the sequence P is zero.

(2) For every integer in sequence N, there exists an integer in sequence P with the same magnitude.

freedom128 · Answer

An increasing arithmetic sequence N = -ve integers only.
An increasing arithmetic sequence P = +ve integers only. 
The sequence C contains all the terms of arithmetic sequences N and P.

Q. Is sequence C an arithmetic sequence?

(1) The sum of the largest term of the sequence N and the smallest term of the sequence P is zero. 
- If N = {-5,-3,-1} and P={1,3,5} then sequence C is an arithmetic sequence
-If N = {-7,-5,-3} and P={3,5,7} then sequence C is NOT an arithmetic sequence. Sequence C lacks of elements {-1, 1} to be qualified as arithmetic sequence. 
NOT SUFFICIENT

(2) For every integer in sequence N, there exists an integer in sequence P with the same magnitude. 
- If N = {-5,-3,-1} and P={1,3,5} then sequence C is an arithmetic sequence
-If N = {-7,-5,-3} and P={3,5,7} then sequence C is NOT an arithmetic sequence. Sequence C lacks of elements {-1, 1} to be qualified as arithmetic sequence. 
NOT SUFFICIENT

Combined (1) and (2) 
- If N = {-5,-3,-1} and P={1,3,5} then sequence C is an arithmetic sequence
-If N = {-7,-5,-3} and P={3,5,7} then sequence C is NOT an arithmetic sequence. Sequence C lacks of elements {-1, 1} to be qualified as arithmetic sequence. 
NOT SUFFICIENT

FINAL ANSWER IS (E)

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exc4libur · Answer

C=AP?

(1) insufic

tnN+aP=0
-1+1=0;  ≠AP
-1+1=0;  =AP

(2) insufic

≠AP
 =AP

(1/2) insufic

≠AP
 =AP

Ans (E)

MayankSingh · Answer

IMO E

N = {N1, N2, N3 ,................., N9 }  
N1-N9 : Negative integers ; Common difference = D (+ve)
P= {P1, P2, ..........................,P9} 
P1-P9 : Positive integers ; Common difference = K (+ve)

(1) The sum of the largest term of the sequence N and the smallest term of the sequence P is zero.
N9+P1=0
N9= - P1

------(N9-D).........N9............0.........N1..........(N1+K)
To be an AP,
D=(N1-N9)=K
So not sufficient

(2) For every integer in sequence N, there exists an integer in sequence P with the same magnitude.
C= { -8,-6,-4 , 4, 6, 8 } - Not in AP
C= {-12,-4,4,12} - In AP
Not sufficient

Together:C= { -8,-6,-4 , 4, 6, 8 } - Not in AP
C= {-12,-4,4,12} - In AP
Not sufficient.

aka007 · Answer

IMO the answer is E

S1: 
largest term on N is -1 & smallest of P is +1 (condition of S1 satisfied)
now, whether C is a AP or not entirely depends on the common difference D. 
If the next term on N is -2 whereas the next term on P is +5. NOT AP
If N (-3,-1) and P (1,3) then yes AP

Hence the statement is not sufficient.

S2:

the terms are of equal magnitude
N(-3,-1) and P (1,3) yes AP
N(-2,-1) and P (1,2) no AP

statement is not sufficient

Combining both the statements we do not get any addl value, therefore the answer is E

Lipun · Answer

Let S(N)=N1+(N-1)C1, N1 is the first term
Let S(P)=P1+(P-1)C2, P1 is the first term

1. Given the largest of S(N) is equal to smallest of S(P)
 No info about the constants. Not sufficient.
 
2. Given for every integer in S(N), there is an integer value in S(P) having same magnitude.
 If C1=4 and C2=2, then also the above statement holds good, but the set S(C) will not be in arithmetic sequence.
 
Considering 1 and 2 together also not sufficient info.

Note: For the overall sequence S(C) to be in arithmetic sequence, largest of S(N) - smallest of S(P) = C1 = C2

Vigasimrair · Answer

the answer is E.
Let's imagine that we have the following 2 sequences:

First case
(N) -10; -8; -6
(P) 6;8;10

or the following 2 sequences:

Second case
(N) -7; -5; -3; -1; 
(P) 1;3;5;7

Both pieces of information are not sufficient, because in the first case we don't have an arithmetic sequence and in the second case we have arithmetic sequence.

debjit1990 · Answer

Ans: B

Statement 1: The sum of the largest term of the sequence N and the smallest term of the sequence P is zero.

N={-9,-7,-5,-3,-1}
P={1,2,3,4,5}
C={-9,-7,-5,-3,-1,1,2,3,4,5}, so C is not an arithmatic sequence
N={-9,-7,-5,-3,-1}
P={1,3,5,7,9}
C={-9,-7,-5,-3,-1,1,3,5,7,9}, so C is an arithmatic sequence

No Statement 1 not sufficient.

(2) For every integer in sequence N, there exists an integer in sequence P with the same magnitude.
N={-9,-7,-5,-3,-1}
P={1,3,5,7,9}
C={-9,-7,-5,-3,-1,1,3,5,7,9}, so C is an arithmatic sequence
Sufficient.

gullyboy09 · Answer

Hi  Bunuel , how come C can have same number of terms as N and P if both N and P have different set of numbers.

egmat · Answer

Hi gullyboy09,

This one is just a wording knot, not a math trap, so let's untangle the sentence itself.

The phrase "the number of terms in sequence C is equal to the number of terms in arithmetic sequences N  and  P" does  not  mean C matches N's count and P's count separately. It means C's count equals the  combined total  of N's terms  plus  P's terms.

In other words, the line is just telling you that C is the simple merge of N and P with  nothing dropped and nothing duplicated :

- count(C) = count(N) + count(P)

The reason they bother to say this is to rule out any overlap. Since every term of N is negative and every term of P is positive, the two sets can never share a number, so when you pour both into C, you keep all of them.

Quick check with tiny numbers: 
- N = {-3, -1} -  2  terms
- P = {1, 3} -  2  terms
- C = {-3, -1, 1, 3} -  4  terms

That  4  is exactly  2  +  2  - the terms of N  and  P, all present. So C always lines up, smallest to largest, as: all of N, then all of P.

Once you read it that way, the actual question becomes clean: C is the full negative-then-positive list, and you're just asking whether that whole list keeps one common difference - which is where statements ( 1 ) and ( 2 ) come in.

Answer: E

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

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