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# M16-26

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Math Expert
Joined: 02 Sep 2009
Posts: 46207

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16 Sep 2014, 00:59
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Difficulty:

35% (medium)

Question Stats:

73% (00:53) correct 27% (01:02) wrong based on 62 sessions

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The sequence $$a_1$$, $$a_2$$, $$a_3$$, ..., $$a_n$$, ... is such that $$a_{n}-a_{n+1}=a_{n+1}$$ for all integers $$n \ge 1$$. What is the value of fourth element of this sequence?

(1) The first element of this sequence is 1

(2) The third element of this sequence is $$\frac{1}{4}$$

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Joined: 02 Sep 2009
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16 Sep 2014, 00:59
Official Solution:

Given that $$a_{n}-a_{n+1}=a_{n+1}$$. So, $$a_{n+1}=\frac{a_{n}}{2}$$.

(1) The first element of this sequence is 1. Given: $$a_1=1$$. Now, since $$a_{n+1}=\frac{a_{n}}{2}$$ then $$a_2=\frac{a_1}{2}=\frac{1}{2}$$, $$a_3=\frac{a_2}{2}=\frac{1}{4}$$ and $$a_4=\frac{a_3}{2}=\frac{1}{8}$$. Sufficient.

(2) The third element of this sequence is $$\frac{1}{4}$$. Given: $$a_3=\frac{1}{4}$$, hence $$a_4=\frac{a_3}{2}=\frac{1}{8}$$. Sufficient.

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Joined: 06 Sep 2012
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25 Mar 2017, 01:01
Hi bunuel please explain how u got an/2.i cudnt undertstand the calculation thank you

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Math Expert
Joined: 02 Sep 2009
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25 Mar 2017, 02:45
matayo25 wrote:
Hi bunuel please explain how u got an/2. i cudnt undertstand the calculation thank you

Posted from my mobile device

Please invest a little bit more time and elaborate your question. Which statement are you talking about?
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25 Mar 2017, 04:04
Hi bunuel i am talking about this part : Given that an-an+1=an+1 so, an+1= an/2 this part i am not able to configure please be kind to let me know how you got an+1=an/2 thank you

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Math Expert
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25 Mar 2017, 06:05
1
1
matayo25 wrote:
Hi bunuel i am talking about this part : Given that an-an+1=an+1 so, an+1= an/2 this part i am not able to configure please be kind to let me know how you got an+1=an/2 thank you

Posted from my mobile device

Given that $$a_{n}-a_{n+1}=a_{n+1}$$.

Add $$a_{n+1}$$ to both parts: $$a_{n}=2*a_{n+1}$$.

So, $$a_{n+1}=\frac{a_{n}}{2}$$.

It's very basic algebraic manipulation. The same as below example:

x - y = y;

y = x/2.

Hope it's clear.
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25 Mar 2017, 06:22
Thanks a lot bunuel

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Re: M16-26   [#permalink] 25 Mar 2017, 06:22
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# M16-26

Moderators: chetan2u, Bunuel

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