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A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by

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A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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A sequence of non-zero terms \(a_1\), \(a_2\), \(a_3\), ..., \(a_{m-1}\), \(a_m\), is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2. If m=12, then how many terms in the given sequence are positive?

(1) \(a_3\) is positive
(2) \(a_4\) is positive

My apologies . I could not find a way to type the sequence here , so I am attaching the DS question .
[Reveal] Spoiler: OA

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Sequence_1.png
Sequence_1.png [ 12.3 KiB | Viewed 2111 times ]


Last edited by Bunuel on 06 Apr 2013, 05:17, edited 1 time in total.
Renamed the topic and edited the question.
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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A sequence of non-zero terms \(a_1\), \(a_2\), \(a_3\), ..., \(a_{m-1}\), \(a_m\), is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2. If m=12, then how many terms in the given sequence are positive?

From above:
\(a_3=(a_2)^2*a_1\);
\(a_4=(a_3)^2*a_2\);
...

(1) \(a_3\) is positive --> \(a_3=(a_2)^2*a_1=positive\) --> \(a_1=positive\). Now, if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive but if \(a_1=1\), and \(a_2=-1\) (\(a_3=(a_2)^2*a_1=(-1)^2*1=1=positive\)), then not all the terms in the sequence will be positive. Not sufficient.

(2) \(a_4\) is positive --> \(a_4=(a_3)^2*a_2=positive\) --> \(a_2=positive\). The same here: if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive but if \(a_1=-1\), and \(a_2=1\) (\(a_3=(a_2)^2*a_1=(1)^2*(-1)=-1\) and \(a_4=(a_3)^2*a_2=(-1)^2*1=1=positive\)), then not all the terms in the sequence will be positive. Not sufficient.

(1)+(2) From above we have that \(a_1=positive\) and \(a_2=positive\). Therefore, all 12 terms of the sequence are positive. Sufficient.

Answer: C.

Hope it's clear.
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 07 Apr 2013, 10:40
Then.. The information for "every k>2" is irrelevant right

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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 07 Apr 2013, 22:33
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 09 Apr 2013, 09:15
Bunuel wrote:
marcovg4 wrote:
Then.. The information for "every k>2" is irrelevant right

Posted from my mobile device Image


"A sequence ... is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2" means that the given formula applies for the terms starting \(a_3\).


Oh I get it, thanks!
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 06 Sep 2013, 12:12
Bunuel wrote:
A sequence of non-zero terms \(a_1\), \(a_2\), \(a_3\), ..., \(a_{m-1}\), \(a_m\), is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2. If m=12, then how many terms in the given sequence are positive?

From above:
\(a_3=(a_2)^2*a_1\);
\(a_4=(a_3)^2*a_2\);
...

(1) \(a_3\) is positive --> \(a_3=(a_2)^2*a_1=positive\) --> \(a_1=positive\). Now, if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive but if \(a_1=1\), and \(a_2=-1\) (\(a_3=(a_2)^2*a_1=(-1)^2*1=1=positive\)), then not all the terms in the sequence will be positive. Not sufficient.

(2) \(a_4\) is positive --> \(a_4=(a_3)^2*a_2=positive\) --> \(a_2=positive\). The same here: if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive but if \(a_1=-1\), and \(a_2=1\) (\(a_3=(a_2)^2*a_1=(1)^2*(-1)=-1\) and \(a_4=(a_3)^2*a_2=(-1)^2*1=1=positive\)), then not all the terms in the sequence will be positive. Not sufficient.

(1)+(2) From above we have that \(a_1=positive\) and \(a_2=positive\). Therefore, all 12 terms of the sequence are positive. Sufficient.

Answer: C.

Hope it's clear.



Although the Answer is correct..but as I see the question Posted and the question in the image are different. Considering the question in the image a1 = +ve, a2=-ve, a3=+ve, a4=-ve and so on...Therefore, there will be 6 +ve terms in the sequence...
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 18 Nov 2013, 11:51
Bunuel wrote:
Now, if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive .


How can we say that because a1 and a2 are positive all terms will be positive?
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 19 Nov 2013, 01:43
nikhil007 wrote:
Bunuel wrote:
Now, if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive .


How can we say that because a1 and a2 are positive all terms will be positive?


\(a_3=(a_2)^2*a_1\);
\(a_4=(a_3)^2*a_2\);
...

Now, if a1 and a2 are both positive can a3 be negative? a4? an?
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 19 Nov 2013, 23:58
guerrero25 wrote:
A sequence of non-zero terms \(a_1\), \(a_2\), \(a_3\), ..., \(a_{m-1}\), \(a_m\), is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2. If m=12, then how many terms in the given sequence are positive?

(1) \(a_3\) is positive
(2) \(a_4\) is positive

My apologies . I could not find a way to type the sequence here , so I am attaching the DS question .


Statement 2 in the question and in the screenshot are different! is \(a_4\) positive or negative?
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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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New post 20 Nov 2013, 01:32
emailmkarthik wrote:
guerrero25 wrote:
A sequence of non-zero terms \(a_1\), \(a_2\), \(a_3\), ..., \(a_{m-1}\), \(a_m\), is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2. If m=12, then how many terms in the given sequence are positive?

(1) \(a_3\) is positive
(2) \(a_4\) is positive

My apologies . I could not find a way to type the sequence here , so I am attaching the DS question .


Statement 2 in the question and in the screenshot are different! is \(a_4\) positive or negative?


The discussion is on the question which says that \(a_4\) is positive.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by [#permalink]

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Re: A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by   [#permalink] 10 Sep 2015, 01:22
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