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# For each positive integer k, let ak = (1 + 1/(k+1)). Is the product a1

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Senior Manager
Joined: 04 Sep 2017
Posts: 291
For each positive integer k, let ak = (1 + 1/(k+1)). Is the product a1  [#permalink]

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21 Sep 2019, 15:20
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Difficulty:

85% (hard)

Question Stats:

46% (02:15) correct 54% (02:26) wrong based on 145 sessions

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For each positive integer k, let $$a_k = (1 + \frac{1}{k+1})$$. Is the product $$a_1a_2 … a_n$$ an integer?

(1) n + 1 is a multiple of 3.
(2) n is a multiple of 2.

DS59851.01
Intern
Joined: 04 Feb 2018
Posts: 44
Re: For each positive integer k, let ak = (1 + 1/(k+1)). Is the product a1  [#permalink]

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23 Sep 2019, 03:16
gmatt1476 wrote:
For each positive integer k, let $$a_k = (1 + \frac{1}{k+1})$$. Is the product $$a_1a_2 … a_n$$ an integer?

(1) n + 1 is a multiple of 3.
(2) n is a multiple of 2.

DS59851.01

The correct answer is B. The detailed working can be found in the attached document.
Attachments

My Solution.doc [40 KiB]

Intern
Joined: 04 Feb 2018
Posts: 44
Re: For each positive integer k, let ak = (1 + 1/(k+1)). Is the product a1  [#permalink]

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23 Sep 2019, 03:33
gmatt1476 wrote:
For each positive integer k, let $$a_k = (1 + \frac{1}{k+1})$$. Is the product $$a_1a_2 … a_n$$ an integer?

(1) n + 1 is a multiple of 3.
(2) n is a multiple of 2.

DS59851.01
coylahood wrote:
gmatt1476 wrote:
For each positive integer k, let $$a_k = (1 + \frac{1}{k+1})$$. Is the product $$a_1a_2 … a_n$$ an integer?

(1) n + 1 is a multiple of 3.
(2) n is a multiple of 2.

DS59851.01

The correct answer is B. The detailed working can be found in the attached document.

Sorry I realised a mistake with my detailed solution. I have uploaded the corrected version.
Attachments

Updated Solution.doc [41 KiB]

Manager
Joined: 01 Feb 2017
Posts: 244
Re: For each positive integer k, let ak = (1 + 1/(k+1)). Is the product a1  [#permalink]

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24 Sep 2019, 02:03
2
for n=1, a1= 3/2
for n=2, a2= 4/3
for n=3, a3= 5/4
for n=4, a4= 6/5

Testing products:
n=even, product is an integer (a1*a2= 2 , a1*a2*a3*a4= 3)
n=odd, product is a non-integer (a1*= 3/2 , a1*a2*a3= 5/2)

St1: n+1 as a multiple of can be either odd or even and hence n can be either as well. Insufficient

St2: n is even. Sufficient.

Ans B
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Joined: 19 Oct 2018
Posts: 1078
Location: India
Re: For each positive integer k, let ak = (1 + 1/(k+1)). Is the product a1  [#permalink]

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18 Oct 2019, 14:23
$$a_k = (1 + \frac{1}{k+1})$$
$$a_k = \frac{k+2}{k+1})$$

$$a_1a_2 … a_n$$= $$\frac{3}{2}*\frac{4}{3}*\frac{5}{4}*.........*\frac{n+2}{n+1}$$

$$a_1a_2 … a_n$$=$$\frac{n+2}{2}$$

$$a_1a_2 … a_n$$= $$\frac{n}{2}+1$$

$$\frac{n}{2}+1$$ is an integer, if $$\frac{n}{2}$$ is an integer

So basically question stem is whether n is a multiple of 2 or not.

Statement 1- n+1 is a multiple of 3

if n+1 is 3, n is 2 (even)
if n+1 is 6, n is 5 (odd)

Insufficient

Statement 2- n is multiple of 2 or even

Sufficient

gmatt1476 wrote:
For each positive integer k, let $$a_k = (1 + \frac{1}{k+1})$$. Is the product $$a_1a_2 … a_n$$ an integer?

(1) n + 1 is a multiple of 3.
(2) n is a multiple of 2.

DS59851.01
Re: For each positive integer k, let ak = (1 + 1/(k+1)). Is the product a1   [#permalink] 18 Oct 2019, 14:23
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