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Math Revolution and GMAT Club Contest! There is a sequence An when n
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05 Dec 2015, 07:52
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Math Revolution and GMAT Club Contest Starts! QUESTION #5:There is a sequence \(A_n\) when n is a positive integer such that \(A_1=a\), \(A_2=b\), and \(A_{n+2}=A_{n+1}*A_n\). Is \(A_6<0\)? (1) a < 0 (2) ab < 0 Check conditions below: Math Revolution and GMAT Club ContestThe Contest Starts November 28th in Quant Forum We are happy to announce a Math Revolution and GMAT Club Contest For the following four (!) weekends we'll be publishing 4 FRESH math questions per weekend (2 on Saturday and 2 on Sunday). To participate, you will have to reply with your best answer/solution to the new questions that will be posted on Saturday and Sunday at 9 AM Pacific. Then a week later, the forum moderator will be selecting 2 winners who provided most correct answers to the questions, along with best solutions. Those winners will get 6months access to GMAT Club Tests. PLUS! Based on the answers and solutions for all the questions published during the project ONE user will be awarded with ONE Grand prize: PS + DS course with 502 videos that is worth $299! All announcements and winnings are final and no whining GMAT Club reserves the rights to modify the terms of this offer at any time. NOTE: Test Prep Experts and Tutors are asked not to participate. We would like to have the members maximize their learning and problem solving process.
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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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05 Dec 2015, 10:09
I will go with B, and this is my reasoning:
There is a sequence An when n is a positive integer such that A1=a, A2=b, and An+2=An+1∗An. Is A6<0?
ok, this means that when n=1, we have A3 = A2 * A1 or A3 = A1 * A2 A4 = A3*A2 A5 = A4*A3 A6 = A5 *A4
We need to find whether A6<0.
(1) a < 0 well, clearly insufficient, since A1 = negative but A2 can be negative or positive: first case: A2 = negative A3 = positive A4 = negative A5 = negative A6 = positive
second case = A2 = positive A3 = negative A4 = negative A5 = positive A6 = negative
since there are 2 outcomes, A is insufficient.
(2) ab < 0 this means that one is positive and the other one is negative. that means that A3 = negative, we thus have 2 cases: A1 negative + A2 positive and A1 positive + A2 negative
1 case: A2 = negative A3 = negative as above mentioned A4 = positive A5 = negative A6 = negative
2nd case: A2 = positive A3 = negative as above mentioned A4 = negative A5 = positive A6 = negative
since both ways get us A6 negative, B is the answer.



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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05 Dec 2015, 10:40
Following information is given:1. An is a sequence and n is a positive integer. 2. A1 = a and A2 = b (for n = 1 and 2 values of the sequence are given) 3. An+2 = An+1∗An (this formula or say pattern is given to find the subsequent values) This pattern tells us that any two consecutive values of the sequence when multiplied give the third consecutive value of the sequence. Hence, Subsequent values of this sequence would be:when n = 3, A3 = A2 * A1 = ba when n = 4, A4 = A3 * A2 = ba*b = b2a when n = 5, A5 = A4 * A3 = b2a*ba = b3a2 when n = 6, A6 = A5 * A4 = b3a2*b2a = b5a3 we are asked if A6 < 0 i.e is A6 negative? i.e. is b5a3 negative?as we see powers of both b and a are odd, b5a3 will be negative if a and b have opposite signs. statement1: a < 0as b can be positive or negative, b5a3 can be positive or negative. As we dont have the definite answer with statement1, its INSUFFICIENT. statement2: ab < 0this tells us that a and b have opposite signs. b5a3 will be negative. We have definite answer with statement2. its SUFFICIENT. Option (B) is correct answer.
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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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11 Dec 2015, 09:55
A1=a, A2=b, and An+2=An+1∗An. Is A6<0?
From equation, An+2=An+1∗An A6=A5*A4 A5=A4*A3 A4=A3*A2 A3=A2*A1
Therefore A6= (A2)^5 * (A1)^3 Now lets take the choices
(1) a < 0 a<0 and b>0, Then A6<0 Yes But if a<0 ,b<0 Then A6 >0 No
Choice 1 is insufficient.
(2) ab < 0 Means that a and b are of opposite signs . Therefore A6 which is equal to (A2)^5 * (A1)^3 will always be negative and A6<0.
Therefore B is sufficient.
Hence choice B is answer.



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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05 Dec 2015, 10:22
The answer to this question is B.
The problem statement  Is A6<0. Based on the question, An = a, b, ab, ab^2, a^2b^3, a^3b^5, ... so A6 = a^3 * b^5. So in order to find if A6 is < 0, we have to find the sign of both a and b, because the power of both a and b is odd and can either be ve or +ve.
1) a<0  here if b is >0, then A6 is ve and if b<0, A6 is +ve (Not Sufficient)
2) ab<0  here there are two possibility a<0 & b>0  in this case, A6 is ve. and a>0 & b<0  in this case also A6 is ve. Hence A6 is <0 when ab <0. (Sufficient).



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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05 Dec 2015, 19:59
A1 = a; A2 = b; A3 = A2 * A1 = a*b A4 = A3 * A2 = ab * b = a * b^2 A5 = A4 * A3 = (a * b^2) * (ab) = (a^2 * b^3) A6 = A5 * A4 = (a^2 * b^3) * (a * b^2) = (a^3 * b^5) = (ab)^3 * b^2
St1: a < 0 > Not sufficient as 'b' is not known
St2: ab < 0 If ab < 0 then a and b is not equal to 0. A6 depends upon the sign of (ab) since b^2 is always positive. So ab < 0 > A6 is negative Statement 2 is sufficient.
Answer: B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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05 Dec 2015, 20:46
I will go with B as my choice of answer.
A1= a, A2=b and An+2= An+1 + An. Question asks us is A6 < 0? this can be re framed as Is a^3 X b^5 < 0?
Case 1: a < 0 this implies a is negative. So a^3 is also negative. This depends on sign of b. If b is positive we will get A6 < 0 but if b is negative then b^5 is also negative so whole term A6 will be positive and that means that A6 may be positive or negative depending on the value of b................... Insufficient
Case 2: ab < 0 this implies either a or b is negative. if a is negative, b will be positive so this would imply A6 < 0 and also if b is negative then a will be positive. So in both case the sign will be preserved and will definitely give us sufficient conditions or answer of question A6 < 0...................... Sufficient
Answer is B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 02:38
From the formula: \(A_6 = A_5 * A_4 = (A_4)^2 * A_3 = (A_3)^3 * (A_2)^2 = (A_2)^5 * (A_1)^3 = b^5*a^3 = (ab)^3 * b^2\) 1. a < 0 is insufficient because \(A_6\) > 0 when b < 0 & \(A_6\) < 0 when b > 0. 2. ab < 0 is sufficient to conclude that \(A_6<0\) because\((ab)^3<0\) and \(b^2>0\) (b cannot equal 0 since ab<0) Hence B.
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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 05:39
There is a sequence An when n is a positive integer such that A1=a, A2=b, and An+2=An+1∗An. Is A6<0?
(1) a < 0 (2) ab < 0
On explanding,
A6 = A1^3 * A2*5 = a^3 *b*5
We are asked if a^3 *b*5 < 0 1) a < 0. Insufficent as we have to know whether b is positive or negative. 2) ab< 0. Sufficient since if ab < 0 then multiplication of add powers of a and odd powers of b will also be less than 0.
Ans. B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 06:26
QUESTION #5:
There is a sequence An when n is a positive integer such that A1=a, A2=b, and An+2=An+1∗An. Is A6<0?
(1) a < 0 (2) ab < 0
Soln: \(A1 = a, A2 = b,\) and since \(An+2=An+1∗An,\) we can deduce \(A3 = A2*A1 = ab.\) Similarly \(A4 = A3*A2 = ab^2, A5 = a^2b^3, and A6 = a^3b^5.\) So, the problem translates to: whether \(a^3b^5 < 0\) ??
1. \(a < 0,\) so, \(a^3 < 0.\) But we do not know b. If b = 1, YES. If b = 1, NO. INSUFFICIENT.
2. \(ab < 0.\) So, \(a & b\) are of opposite sign. If \(a<0\) and \(b>0,\) then \(a^3<0\) and \(b^5>0,\) and so \(A6 < 0.\) YES. If \(a>0\) and \(b<0,\) then \(a^3>0\) and \(b^5<0,\) and so \(A6 < 0.\) YES. Both YES. SUFFICIENT.
Answer: B.



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 08:50
A1=a,A2=b
An+2=An+1*An
suppose n=1, then A3=A2*A1=ab
A4=A3*A2=ab*b=ab^2
A5=A4*A3=ab^2*ab=a^2b^3
A6=A5*A4=a^2b^3*ab^2=a^3b^5<0.
from above both a and b have odd powers .
Statement 1 if a<0, we cant conclude we don't know value of b
Statement 2 ab<0 means I a>0 and b<0 or a<0 and b>0. both value satisfy above A6<0. so option B is correct



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 09:40
given, \(A_1\) = a \(A_2\) = b
from the formula, \(A_{n+2}\) = \(A_{n+1}\)* A, we get the remaining terms. \(A_3\) = \(A_2\)* \(A_1\) = a*b = ab \(A_4\) = \(A_3\)* \(A_2\) = ab * b = a\(b^2\) \(A_5\) = \(A_4\)* \(A_3\) = a\(b^2\)* ab = \(a^2\)\(b^3\) \(A_6\) = \(A_5\)* \(A_4\) = \(a^2\)\(b^3\)* a\(b^2\) = \(a^3\)\(b^5\)
Now, we need to know if \(a^3\)\(b^5\) is negative or not since, the powers of a and b are odd, we have 3 cases. The product can be 1. positive ( if both a & b are of same sign.) 2. negative ( if one of them is negative and other positive) 3. zero. (if any one or both of them are zero)
Stmt 1: a <0 It doesn't give any information about b.  b can be positive  in that case, the product will be negative.  b can be negative in that case, the product will be positive.  b can be zero  in that case, the product will be zero. Hence we cant conclude using the statement 1 alone.
stmt 2: ab <0 This is possible only if one of them is negative and the other is positive. Also, it is to be noted that none of them is zero. Hence we can conclude that the product \(a^3\)\(b^5\) will be negative.
Answer: B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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Updated on: 12 Dec 2015, 13:39
We have a series where every term from the third term onwards is determined by multiplying the previous two terms. If either or both of the previous two terms are 0 then that term (and every following term) will be 0. If the previous two terms are both the same sign then the term we are calculating will be +ve, if the two previous terms are different in sign then the term we are calculating will be ve.
Since we're looking to determine whether A6<0 based on information about whether a<0 or the product ab<0 let's quickly jot out all of the possible sequences:
ab....A6 ++++++ ++ ++ ++ 000000 ?00000 0?0000
Condition 1) tells us that a is ve, which means that the highlighted sequences are possible. Since in the highlighted sequences term A6 can be +ve, ve or 0, 1) on it's own is not sufficient, we can rule out answers A and D.
Condition 2) tells us that the product ab<0. This means that neither a nor b is 0, and that a and b are of different signs. Therefore the bold sequences are possible. Both of the bold sequences have a ve A6, therefore 2) is sufficient to answer the question "is A6<0", and the correct answer is B.
Originally posted by arthearoth on 06 Dec 2015, 10:09.
Last edited by arthearoth on 12 Dec 2015, 13:39, edited 1 time in total.



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 11:40
A1=a, A2=b, and An+2=An+1∗An.
A6 = A(4+2) = A(4+1).A4
A3= A(1+2) =A2.A1 = ab
A4= A(2+2) = A3.A2 = ab^2
A5= A(3+2) = A4.A3 = a^2b^3
A6= A(4+2) = A5.A4 = a^3b^5 = (ab)^3 b^2..
1) a<0, b+? NS 2) ab<0 .. A6 <0 S
B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 17:16
Quote: There is a sequence \(A_n\) when n is a positive integer such that \(A_1\)=a, \(A_2\)=b, and \(A_n+2\)=\(A_n+1∗A_n\). Is \(A_6\)<0?
(1) a < 0 (2) ab < 0 \(A_3 = A_2*A_1 = b*a\) \(A_4 = A_3*A_2 = (b*a)*b\) \(A_5 = A_4*A_3 = ((b*a)*b)*(b*a)\) \(A_6 = A_5*A_4 = ((b*a)*b)*(b*a)*(b*a)*b = (b*a)^3*b^2\) Statement (1): a < 0 Since we don't have any info on b statement (1) is insufficient Statement (2): ab < 0 \(A_6 = (b*a)^3*b^2\) is < 0 Sufficient Answer (B)



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 19:16
A1= a A2 = b A3= ab A4= ab^2 A5=a^2b^3 A6 = a^3b^5
from statement 1: a<0 we do not know about b is b <0 then a6<0 if b>0 the a6 is not less than 0
in sufficient
statement 2: ab<0
A6= a^3 B^5 this can be written as a^2B^4 * ab a^2B^4 will always be positive since ab< 0 A6 < 0
Sufficient Ans: B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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06 Dec 2015, 19:44
A3 = A1*A2 , A4 = A3*A2 => A4 = A1*A2^2 , A5 = A4*A3 => A5 = A1^2 * A2^3 ly A6 = A1^3 * A2^5 We have to check if (a^3 * b^5) < 0 1) a<0 but we have no info about b, thus insufficient
2) ab<0 which means either a<0 or b<0 thus A6<0 thus (a^3 * b^5) < 0 . Therefore sufficient
Answer is B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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07 Dec 2015, 04:53
A6=A5*A4 =A4*A3*A3*A2 =A3*A2*A2*A1*A2*A1*A2 =A2*A1*A2*A2*A1*A2*A1*A2 =b*a*b*b*a*b*a*b =\(b^5*a^3\)
Since both the powers are odd, we only need to find out whether the a*b with odd powers is positive or negative.
1. a<0 Nothing for b Not Sufficient
2. ab<0 ab with odd powers is given here. Sufficient
Answer: B



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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07 Dec 2015, 05:22
There is a sequence An when n is a positive integer such that A1=a, A2=b, and An+2=An+1∗An. Is A6<0?
(1) a < 0 (2) ab < 0
A6 = a^3 b^5
(1) Not sufficient as dont know anything about the sign of b (2) if ab<0 ; then either of the two is negative. That is, both has opposite sign and that is what we need since we have odd powers of a and b in A6. Hence sufficient.
B.



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Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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07 Dec 2015, 05:50
OA: B
Solution: A1 = a A2 = b A3 = a * b A4 = a * b^2 A5 = a^2 * b^3 A6 = a^3 * b ^ 5
For A6 to be < 0, either a < 0 (and b > 0) or vice versa
(1) a < 0 No info about b, so insufficient. (2) ab < 0 this is exactly the condition we need as stated above, hence sufficient.




Re: Math Revolution and GMAT Club Contest! There is a sequence An when n
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