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# A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the

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A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the  [#permalink]

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29 Mar 2007, 18:53
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A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the following is in the sequence?

A. 105
B. 786
C. 966
D. 1025
Veritas Prep GMAT Instructor
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Location: Pune, India
Re: Sequence question  [#permalink]

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19 Jan 2011, 20:30
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A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence?

105
786
966
1025

Given:
$$a_1 = 64$$
$$a_2 = 66$$
$$a_3 = 67$$
...
$$a_n = 8 + a_{n - 3}$$

So $$a_4 = 8 + a_1$$ = 8 + 64
$$a_5 = 8 + a_2$$ = 8 + 66
$$a_6 = 8 + a_3$$ = 8 + 67
$$a_7 = 8 + a_4$$ = 8 + 8 + 64
$$a_8 = 8 + a_5$$ = 8 + 8 + 66
and so on...

So any number that belongs to this sequence will be sum of one of 64/66/67 and some number of 8s.

105 - 64 = 41 which is not a multiple of 8. 41 is 1 more than a multiple of 8 so when you subtract 2/3 out of 41 (in effect subtracting 66/67 out of 105), you will still not get a multiple of 8. Hence 105 is not in this sequence.

786 - 64 = 720 which is divisible by 8 hence it will be in the sequence. This is your answer and ideally you should stop here. But if you want to check the remaining two options:

966 - 64 = 902 which is not divisible by 8. Neither are 900 and 899. Or say that 902 is 6 more than a multiple of 8 so when you subtract 2/3 out of it, you will still not get a multiple of 8.

1025 - 64 = 961 which is not divisible by 8 and is 1 more than a multiple of 8 so when you subtract 2/3 out of it, it will still not give a multiple of 8.
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Karishma
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Save up to $1,000 on GMAT prep through 8/20! Learn more here > GMAT self-study has never been more personalized or more fun. Try ORION Free! Math Expert Joined: 02 Sep 2009 Posts: 47923 Re: Sequence question [#permalink] ### Show Tags 21 Jan 2011, 13:18 6 1 bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. Bhanu Easier way would be to write down several terms from the sequence: $$a_1 = 64$$ $$a_2 = 66$$ $$a_3 = 67$$ $$a_4 = 8 + a_1 = 72$$ $$a_5 = 8 + a_2 = 74$$ $$a_6 = 8 + a_3 = 75$$ ... $$a_n = 8 + a_{n - 3}$$ Note that the terms in the sequence have remainder of 0 ($$a_1$$, $$a_4$$, $$a_7$$, ...), 2 ($$a_2$$, $$a_5$$, $$a_8$$, ...) or 3 ($$a_3$$, $$a_6$$, $$a_9$$, ...) upon division by 8. Only 786 has appropriate remainder of 2 (105 and 1025 has a remainder of 1 upon division by 8 and 966 has a remainder of 6). Answer: B. Hope it's clear. _________________ ##### General Discussion Manager Joined: 28 Feb 2007 Posts: 189 Location: California Re: Sequence question [#permalink] ### Show Tags 29 Mar 2007, 20:07 bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. :thanks Bhanu 786 Take each number subtract 64, 66 and 67 and test which of the remainders is a multiple of 8. Senior Manager Joined: 20 Feb 2007 Posts: 256 ### Show Tags 01 Apr 2007, 21:54 an=8+an-3 Is this an=8+(an)-3 OR an=8+a(n-3)??? Intern Joined: 19 Jul 2009 Posts: 45 Location: baltimore, md Schools: kellogg, booth, stern, ann arbor Re: Sequence question [#permalink] ### Show Tags 27 Sep 2009, 13:48 techjanson wrote: bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. Bhanu 786 Take each number subtract 64, 66 and 67 and test which of the remainders is a multiple of 8. Is anyone sure if this is correct? i'm getting that 786 AND 966 would both be in the sequence. here is my reasoning: the question stem gives "an=8+an-3" so: 64+8-3 = 69 66+8-3 = 71 67+8-3 = 72 In fact, we could simply each by just adding 5 to each subsequent number and we start seeing a pattern... 64, 66, 67,69, 71, 72, 74, 76, 77, 79,81, (Please note the color scheme here) You will notice that each number increases by 5 and always follows a units digit pattern. 64....9,4,9,4,9,4 66....1,6,1,6,1,6 77....2,7,2,7,2,7 so whatever the answer is, it must have a units digit that follows this pattern. Only 786 and 966 do. They follow the 66....1,6,1,6,1,6 pattern. furthermore, it should always follow...66, 71,76,81,86,91,96,101,106.........786.....966. anyone able to verify the answer? Or maybe my reasoning is flawed somewhere. if anyone knows, please say. Thanks _________________ Paaaaayyy Meeeee!!!!! Manager Joined: 25 Feb 2009 Posts: 55 Re: Sequence question [#permalink] ### Show Tags 15 Oct 2009, 09:13 azule45 wrote: techjanson wrote: bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. Bhanu 786 Take each number subtract 64, 66 and 67 and test which of the remainders is a multiple of 8. Is anyone sure if this is correct? i'm getting that 786 AND 966 would both be in the sequence. here is my reasoning: the question stem gives "an=8+an-3" so: 64+8-3 = 69 66+8-3 = 71 67+8-3 = 72 In fact, we could simply each by just adding 5 to each subsequent number and we start seeing a pattern... 64, 66, 67,69, 71, 72, 74, 76, 77, 79,81, (Please note the color scheme here) You will notice that each number increases by 5 and always follows a units digit pattern. 64....9,4,9,4,9,4 66....1,6,1,6,1,6 77....2,7,2,7,2,7 so whatever the answer is, it must have a units digit that follows this pattern. Only 786 and 966 do. They follow the 66....1,6,1,6,1,6 pattern. furthermore, it should always follow...66, 71,76,81,86,91,96,101,106.........786.....966. anyone able to verify the answer? Or maybe my reasoning is flawed somewhere. if anyone knows, please say. Thanks I suppose the question is : An = 8 + A (n-3) eg. A4 = 8 + A1; A5 = 8 + A2.... Thanks techjason, I take your point now. Director Joined: 23 Apr 2010 Posts: 553 Re: Sequence question [#permalink] ### Show Tags 24 Jan 2011, 01:54 Thanks, Bunuel. It helps. Manager Joined: 08 Sep 2010 Posts: 134 Re: Sequence question [#permalink] ### Show Tags 25 Jan 2011, 20:46 Thanks Karishma for the explanation. _________________ My will shall shape the future. Whether I fail or succeed shall be no man's doing but my own. If you like my explanations award kudos. Intern Joined: 16 Nov 2010 Posts: 18 Re: Sequence question [#permalink] ### Show Tags 07 Mar 2011, 04:32 Bunuel wrote: bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. Bhanu Easier way would be to write down several terms from the sequence: $$a_1 = 64$$ $$a_2 = 66$$ $$a_3 = 67$$ $$a_4 = 8 + a_1 = 72$$ $$a_5 = 8 + a_2 = 74$$ $$a_6 = 8 + a_3 = 75$$ ... $$a_n = 8 + a_{n - 3}$$ Note that the terms in the sequence have remainder of 0 ($$a_1$$, $$a_4$$, $$a_7$$, ...), 2 ($$a_2$$, $$a_5$$, $$a_8$$, ...) or 3 ($$a_3$$, $$a_6$$, $$a_9$$, ...) upon division by 8. Only 786 has appropriate remainder of 2 (105 and 1025 has a remainder of 1 upon division by 8 and 966 has a remainder of 6). Hope it's clear. Hi, Could you pls explain the underlined part i.e why do we need to find that out? Thanks. Math Expert Joined: 02 Sep 2009 Posts: 47923 Re: Sequence question [#permalink] ### Show Tags 07 Mar 2011, 06:28 1 deepaksharma1986 wrote: Bunuel wrote: bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. Bhanu Easier way would be to write down several terms from the sequence: $$a_1 = 64$$ $$a_2 = 66$$ $$a_3 = 67$$ $$a_4 = 8 + a_1 = 72$$ $$a_5 = 8 + a_2 = 74$$ $$a_6 = 8 + a_3 = 75$$ ... $$a_n = 8 + a_{n - 3}$$ Note that the terms in the sequence have remainder of 0 ($$a_1$$, $$a_4$$, $$a_7$$, ...), 2 ($$a_2$$, $$a_5$$, $$a_8$$, ...) or 3 ($$a_3$$, $$a_6$$, $$a_9$$, ...) upon division by 8. Only 786 has appropriate remainder of 2 (105 and 1025 has a remainder of 1 upon division by 8 and 966 has a remainder of 6). Hope it's clear. Hi, Could you pls explain the underlined part i.e why do we need to find that out? Thanks. Because it helps to find the answer... Numbers in the sequence can have only 3 remainders upon division by 8: 0, 2, or 3. Among the answer choices only 786 has appropriate remainder of 2 thus only 786 can be in the sequence. _________________ Manager Joined: 14 Feb 2011 Posts: 180 Re: Sequence question [#permalink] ### Show Tags 07 Mar 2011, 07:39 deepaksharma1986 wrote: Bunuel wrote: bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. Bhanu Easier way would be to write down several terms from the sequence: $$a_1 = 64$$ $$a_2 = 66$$ $$a_3 = 67$$ $$a_4 = 8 + a_1 = 72$$ $$a_5 = 8 + a_2 = 74$$ $$a_6 = 8 + a_3 = 75$$ ... $$a_n = 8 + a_{n - 3}$$ Note that the terms in the sequence have remainder of 0 ($$a_1$$, $$a_4$$, $$a_7$$, ...), 2 ($$a_2$$, $$a_5$$, $$a_8$$, ...) or 3 ($$a_3$$, $$a_6$$, $$a_9$$, ...) upon division by 8. Only 786 has appropriate remainder of 2 (105 and 1025 has a remainder of 1 upon division by 8 and 966 has a remainder of 6). Hope it's clear. Hi, Could you pls explain the underlined part i.e why do we need to find that out? Thanks. The need to find this out comes from understanding the fact that from 4th term onwards, all the terms are of one of the three forms namely 8k+64 or 8k+66 or 8k+67, as a4 is a1+8 and so on. Therefore, we can deduce an important characteristic that any term of the sequence when divided by 8 should have remainder 0 or 2 or 3 and use this deduction to eliminate incorrect choices. Math Expert Joined: 02 Sep 2009 Posts: 47923 Re: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the [#permalink] ### Show Tags 09 Jul 2013, 09:31 Bumping for review and further discussion. _________________ Intern Joined: 28 May 2012 Posts: 26 Concentration: Finance, General Management GMAT 1: 700 Q50 V35 GPA: 3.28 WE: Analyst (Investment Banking) Re: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the [#permalink] ### Show Tags 09 Jul 2013, 19:37 2 Here is my idea: a1 = 64 = 8*8= 8*k1 a4 = a1 +8 = 8*8 +8 = 8*k4 a7 = a4 +8 = 8*k4 + 8 = 8*k7 ... a(n)=8*k(n) Similarly, a(2)=66 = 8*k1 +2 -> a(2)-2 = 8*k(1) a(5)-2 = [a(2) - 2] + 8 = 8*k(1) +8 = 8*k(2) --> a(n') -2 = 8*k(n) We apply trial and error to each number, if x, x-2 or x-3 is divisible by 8, it would be the answer. Intern Joined: 21 May 2013 Posts: 28 Location: India Concentration: Finance, Marketing GMAT 1: 660 Q49 V32 Re: Sequence question [#permalink] ### Show Tags 09 Aug 2013, 05:48 Bunuel wrote: bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 can any one help me with this. Bhanu Easier way would be to write down several terms from the sequence: $$a_1 = 64$$ $$a_2 = 66$$ $$a_3 = 67$$ $$a_4 = 8 + a_1 = 72$$ $$a_5 = 8 + a_2 = 74$$ $$a_6 = 8 + a_3 = 75$$ ... $$a_n = 8 + a_{n - 3}$$ Note that the terms in the sequence have remainder of 0 ($$a_1$$, $$a_4$$, $$a_7$$, ...), 2 ($$a_2$$, $$a_5$$, $$a_8$$, ...) or 3 ($$a_3$$, $$a_6$$, $$a_9$$, ...) upon division by 8. Only 786 has appropriate remainder of 2 (105 and 1025 has a remainder of 1 upon division by 8 and 966 has a remainder of 6). Answer: B. Hope it's clear. thank you Bunuel for explaining it.. SVP Joined: 06 Sep 2013 Posts: 1851 Concentration: Finance Re: Sequence question [#permalink] ### Show Tags 18 Dec 2013, 15:38 VeritasPrepKarishma wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 Given: $$a_1 = 64$$ $$a_2 = 66$$ $$a_3 = 67$$ ... $$a_n = 8 + a_{n - 3}$$ So $$a_4 = 8 + a_1$$ = 8 + 64 $$a_5 = 8 + a_2$$ = 8 + 66 $$a_6 = 8 + a_3$$ = 8 + 67 $$a_7 = 8 + a_4$$ = 8 + 8 + 64 $$a_8 = 8 + a_5$$ = 8 + 8 + 66 and so on... So any number that belongs to this sequence will be sum of one of 64/66/67 and some number of 8s. 105 - 64 = 41 which is not a multiple of 8. 41 is 1 more than a multiple of 8 so when you subtract 2/3 out of 41 (in effect subtracting 66/67 out of 105), you will still not get a multiple of 8. Hence 105 is not in this sequence. 786 - 64 = 720 which is divisible by 8 hence it will be in the sequence. This is your answer and ideally you should stop here. But if you want to check the remaining two options: 966 - 64 = 902 which is not divisible by 8. Neither are 900 and 899. Or say that 902 is 6 more than a multiple of 8 so when you subtract 2/3 out of it, you will still not get a multiple of 8. 1025 - 64 = 961 which is not divisible by 8 and is 1 more than a multiple of 8 so when you subtract 2/3 out of it, it will still not give a multiple of 8. Followed the same approach, the only problem is that 786 - 64 is NOT 720. Therefore, neither answer choice works Please advice Cheers! J Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8187 Location: Pune, India Re: Sequence question [#permalink] ### Show Tags 18 Dec 2013, 19:59 jlgdr wrote: VeritasPrepKarishma wrote: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence? 105 786 966 1025 Given: $$a_1 = 64$$ $$a_2 = 66$$ $$a_3 = 67$$ ... $$a_n = 8 + a_{n - 3}$$ So $$a_4 = 8 + a_1$$ = 8 + 64 $$a_5 = 8 + a_2$$ = 8 + 66 $$a_6 = 8 + a_3$$ = 8 + 67 $$a_7 = 8 + a_4$$ = 8 + 8 + 64 $$a_8 = 8 + a_5$$ = 8 + 8 + 66 and so on... So any number that belongs to this sequence will be sum of one of 64/66/67 and some number of 8s. 105 - 64 = 41 which is not a multiple of 8. 41 is 1 more than a multiple of 8 so when you subtract 2/3 out of 41 (in effect subtracting 66/67 out of 105), you will still not get a multiple of 8. Hence 105 is not in this sequence. 786 - 64 = 720 which is divisible by 8 hence it will be in the sequence. This is your answer and ideally you should stop here. But if you want to check the remaining two options: 966 - 64 = 902 which is not divisible by 8. Neither are 900 and 899. Or say that 902 is 6 more than a multiple of 8 so when you subtract 2/3 out of it, you will still not get a multiple of 8. 1025 - 64 = 961 which is not divisible by 8 and is 1 more than a multiple of 8 so when you subtract 2/3 out of it, it will still not give a multiple of 8. Followed the same approach, the only problem is that 786 - 64 is NOT 720. Therefore, neither answer choice works Please advice Cheers! J Yes, that's right. But when you check by subtracting another 2 (to account for 66), you get 720, a multiple of 8. _________________ Karishma Veritas Prep GMAT Instructor Save up to$1,000 on GMAT prep through 8/20! Learn more here >

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Re: A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the  [#permalink]

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13 Jan 2014, 05:48
Given $$a_n=8+a_{(n-3)}$$
=>
$$a_4 = 8+a_1 = 72$$
$$a_5 = 8+a_2 = 74$$
$$a_6 = 8+a_3 = 75$$

so the sequence is 64,66,67, 72,74,75... meaning each number in the sequence when divided by 8 leaves remainder of either 0 or 2 or 3.

Out of the choices only 786 leaves 2 as remainder. hence answer is B
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Re: Sequence question  [#permalink]

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06 Feb 2014, 11:12
VeritasPrepKarishma wrote:
jlgdr wrote:
VeritasPrepKarishma wrote:
A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence?

105
786
966
1025

Given:
$$a_1 = 64$$
$$a_2 = 66$$
$$a_3 = 67$$
...
$$a_n = 8 + a_{n - 3}$$

So $$a_4 = 8 + a_1$$ = 8 + 64
$$a_5 = 8 + a_2$$ = 8 + 66
$$a_6 = 8 + a_3$$ = 8 + 67
$$a_7 = 8 + a_4$$ = 8 + 8 + 64
$$a_8 = 8 + a_5$$ = 8 + 8 + 66
and so on...

So any number that belongs to this sequence will be sum of one of 64/66/67 and some number of 8s.

105 - 64 = 41 which is not a multiple of 8. 41 is 1 more than a multiple of 8 so when you subtract 2/3 out of 41 (in effect subtracting 66/67 out of 105), you will still not get a multiple of 8. Hence 105 is not in this sequence.

786 - 64 = 720 which is divisible by 8 hence it will be in the sequence. This is your answer and ideally you should stop here. But if you want to check the remaining two options:

966 - 64 = 902 which is not divisible by 8. Neither are 900 and 899. Or say that 902 is 6 more than a multiple of 8 so when you subtract 2/3 out of it, you will still not get a multiple of 8.

1025 - 64 = 961 which is not divisible by 8 and is 1 more than a multiple of 8 so when you subtract 2/3 out of it, it will still not give a multiple of 8.

Followed the same approach, the only problem is that 786 - 64 is NOT 720. Therefore, neither answer choice works

Cheers!
J

Yes, that's right. But when you check by subtracting another 2 (to account for 66), you get 720, a multiple of 8.

Yes actually what I did is notice that we had three cases right?

64 + 8k

66 + 8k

67 + 8k

Now, the first one is always a multiple of 8, the second one is a multiple of 8 plus 2, and the third one is a multiple of 8 plus 3

So we need to find the answer choice that fits the bill

Only B does, being a multiple + 2.

Hence the correct answer

Cheers
J
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Joined: 13 Dec 2013
Posts: 38
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Re: A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the  [#permalink]

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15 Jul 2014, 13:00
Calculate:

a(4) = 72
a(5) = 74
a(6) = 75

That's when you discover the trick: it is always a multiple of 8 or a multiple of 8+2 or +3.
So just check for 3 things:

1) Is it a multiple of 8?
2) Is it 2 more than a multiple of 8?
3) Is it 3 more than multiple of 8?

a) 105:
Closest multiple of 8 = 104 but 105 is only 1 more than a multiple so it is not an answer

b) 786:
Closest multiple of 8 = 8*100 - 8*2 = 784
784 + 2 = 786 so it will be obtained in the sequence.

Hope it helps!
Re: A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the &nbs [#permalink] 15 Jul 2014, 13:00

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