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The next number in a certain sequence is defined by multiply

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The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

(1) The ninth term in this sequence is 81.

(2) The fifth term in this sequence is 1
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The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\) and \(a_4\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.
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Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\), \(a_4\), and \(a_5\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


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Hi All,

This is great 'concept' question. With a bit of 'playing around' you can probably find the patterns involved.

This DS question tells us that each term in a sequence is equal to the PREVIOUS term multiplied by a POSITIVE CONSTANT.

For example, the sequence 1, 2, 4, 8, 16 would fit this definition (another example would be 16, 8, 4, 2, 1, 1/2, etc.). We don't know any of the terms though and we don't know the constant (it could be either an integer, fraction or mixed number). We DO know that since we're multiplying by a positive constant that the sequence of numbers either "increases" or "decreases."

We're asked how many of the first 9 terms are greater than 1? It's interesting that the question asks how many are greater than 1.......

Fact 1: The 9th term is 81

Since we don't know what the constant "K" is (it could be a positive fraction or a positive number > 1), we can TEST VALUES.

IF...
K = 3, then the terms (working backwards from the 9th term....) are:
81, 27, 9, 3, 1, 1/3, 1/9, 1/27, 1/81
Here, the number of terms greater than 1 is 4

IF...
K = 1/3, then the terms (working backwards from the 9th terms....) are:
81, 243, 729, then they get bigger and bigger.....
Here, the number of terms greater than 1 is 9
Fact 1 is INSUFFICIENT.

Fact 2: The fifth term is 1.

Since the sequence either increases or decreases, we'd have...

4 numbers less than 1, then the number 1, then 4 numbers greater than 1

OR

4 numbers greater than 1, then the number 1, then 4 numbers less than 1

Regardless of which option, we end up with exactly 4 terms that are greater than 1.
Fact 2 is SUFFICIENT

Final Answer:
[Reveal] Spoiler:
B


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Re: The next number in a certain sequence is defined by multiply [#permalink]

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New post 04 Jul 2016, 06:33
What if the first term is 1 ? should not c be the answer?

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The next number in a certain sequence is defined by multiply [#permalink]

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New post 03 Dec 2016, 19:06
Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\), \(a_4\), and \(a_5\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


Just wondering,

can K take any positive fractional value or does it need to be an interger.

It does not explicitly mentions that K has to be an integer. I agree the answer will be the same in both the cases however just want to clarify in case we face a similar situation during the main exam.

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New post 04 Dec 2016, 12:22
Hi AmritaSarkar89,

In my solution (above), I discuss this exact issue. GMAT questions are very carefully worded, so you have to pay attention to the details (what those details state and DON'T state). Here, we're told that K is a POSITIVE CONSTANT. That's does not mean that K is necessarily an integer (in the same way that stating N > 0 does not mean that N is necessarily an integer). Be mindful of these details when working through GMAT questions and make sure to write everything on the pad when you're working (so that you don't forget anything).

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Re: The next number in a certain sequence is defined by multiply [#permalink]

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New post 04 Dec 2016, 21:11
EMPOWERgmatRichC wrote:
Hi AmritaSarkar89,

In my solution (above), I discuss this exact issue. GMAT questions are very carefully worded, so you have to pay attention to the details (what those details state and DON'T state). Here, we're told that K is a POSITIVE CONSTANT. That's does not mean that K is necessarily an integer (in the same way that stating N > 0 does not mean that N is necessarily an integer). Be mindful of these details when working through GMAT questions and make sure to write everything on the pad when you're working (so that you don't forget anything).

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Thanks a lot for the detailed reply

Will surely keep in mind :)

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New post 05 Dec 2016, 03:10
But what if a1=-1 and K=2? or a1=0?

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New post 05 Dec 2016, 11:06
Hi ldelcerro,

While it's certainly beneficial to think about all the possibilities (including negatives and 0), neither of those options 'fits' the given information in the two Facts. If the 9th term is 81, then the first term cannot be negative and it cannot be 0 (since repeatedly multiplying either of those options by a positive constant will NEVER lead to +81). The same issue occurs if the 5th term is 1. Thus, that initial term MUST be a positive.

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The next number in a certain sequence is defined by multiply [#permalink]

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New post 19 Apr 2017, 17:51
What I don't understand about this question is for statement 1 why can we not have a sequence such as k=1/2 and

A[16, -8, 4, -2 , 1, -1/2 , 1/4, 1/8, 1/-16] ?


Actually nvm in that instance the number of integers greater than 1 would still be the same if we kept K and made all the values positive.

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The next number in a certain sequence is defined by multiply [#permalink]

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New post 04 Jun 2017, 01:04
Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\), \(a_4\), and \(a_5\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


Hi Bunuel,

Could you help to elaborate the highlighted part by giving an example to illustrate?
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Re: The next number in a certain sequence is defined by multiply [#permalink]

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New post 04 Jun 2017, 05:07
hazelnut wrote:
Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\), \(a_4\), and \(a_5\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


Hi Bunuel,

Could you help to elaborate the highlighted part by giving an example to illustrate?


If \(a_1=1\), then from \(a_9=a_1*k^8=81\) --> \(1*k^8=81\) --> \(k=\sqrt{3}\) --> \(a_2=a_1*k=\sqrt{3}>1\) and so on.

If \(a_1=2\), then from \(a_9=a_1*k^8=81\) --> \(2*k^8=81\) --> \(k \approx 1.6\) --> \(a_2=a_1*k \approx 3.3 > 1\) and so on.
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The next number in a certain sequence is defined by multiply [#permalink]

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New post 04 Jun 2017, 16:37
Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\), \(a_4\), and \(a_5\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


Hi Bunuel, I think \(a_5\) should not count in the term as in highlighted part since \(a_5=1\). If \(a_1>1\), \(a_5=1\), then (\(a_1\), \(a_2\), \(a_3\), \(a_4\)) will be > 1.

Quote:
If \(a_1=2\), then from \(a_9=a_1*k^8=81\) --> \(2*k^8=81\) --> \(k \approx 1.6\) --> \(a_2=a_1*k \approx 3.3 > 1\) and so on.


How could we calculate the value of k without using the calculator? \(2*k^8=81\) --> \(k^8=40.5\) --> \(k \approx 1.6\)
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The next number in a certain sequence is defined by multiply [#permalink]

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New post 04 Jun 2017, 22:04
hazelnut wrote:
Quote:
If \(a_1=2\), then from \(a_9=a_1*k^8=81\) --> \(2*k^8=81\) --> \(k \approx 1.6\) --> \(a_2=a_1*k \approx 3.3 > 1\) and so on.


How could we calculate the value of k without using the calculator? \(2*k^8=81\) --> \(k^8=40.5\) --> \(k \approx 1.6\)


We don't need exact value. From \(2*k^8=81\) it should be clear that k will be greater than 1, which is basically what we want to know. All further conclusions could be made based only on this.
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Re: The next number in a certain sequence is defined by multiply [#permalink]

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New post 05 Jun 2017, 03:29
Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\) and \(a_4\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


How do we know that k is an integer. if k is a fraction then statement 2 does not become insufficient? Bunuel please respond.

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Re: The next number in a certain sequence is defined by multiply [#permalink]

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New post 05 Jun 2017, 06:01
sonikavadhera wrote:
Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\) and \(a_4\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


How do we know that k is an integer. if k is a fraction then statement 2 does not become insufficient? Bunuel please respond.


Your question is not clear. First of all, we do not assume that k is an integer. Next, (2) IS sufficient. Please re-read the solution.
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Re: The next number in a certain sequence is defined by multiply [#permalink]

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New post 05 Jun 2017, 10:39
Bunuel wrote:
sonikavadhera wrote:
Bunuel wrote:
The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1. How many of the first nine terms in this sequence are greater than 1?

Given that:
\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k>0 and n=9.

(1) The ninth term in this sequence is 81 --> \(a_9=a_1*k^8=81\). If \(a_1=1\), then all but \(a_1\) will be greater than 1, but if \(a_1=2\), then all will be greater than 1. Not sufficient.

(2) The fifth term in this sequence is 1 --> \(a_5=a_1*k^4=1\). Now, if \(a_1<1\), then \(k>1\), and all terms from \(a_5\) (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)), so 4 terms will be greater than 1 AND if \(a_1>1\), then \(k<1\), and all terms till \(a_5\) (\(a_2\), \(a_3\) and \(a_4\)), so again 4 terms will be greater than 1. Sufficient.

Answer: B.


How do we know that k is an integer. if k is a fraction then statement 2 does not become insufficient? Bunuel please respond.


Your question is not clear. First of all, we do not assume that k is an integer. Next, (2) IS sufficient. Please re-read the solution.


I am not sure how statement 2 is sufficient if k is a fraction.?

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Re: The next number in a certain sequence is defined by multiply [#permalink]

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New post 05 Jun 2017, 18:37
Hi banuel,
I tried this problem in different way..
1. given 9th term is 81 .. lets find factors of 81 -> 3*3*3*3, so it could be 27*3 or 9*9
lets start with k as 3, then a5 term is 1 means we have 4 terms above 1
lets take k as 9, then a7 is 1, we have 2 terms above 1.. so 1 is NS

2. a5 is 1, a9 = a5*3*3*3*3 = 81, means k is 3, and B is sufficient

ans B

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Re: The next number in a certain sequence is defined by multiply   [#permalink] 05 Jun 2017, 18:37

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