Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 18 Feb 2007
Posts: 4

A sequence, a1=64, a2=66, a3=67, an=8+a(n3), which of the
[#permalink]
Show Tags
29 Mar 2007, 18:53
Question Stats:
62% (02:31) correct 38% (02:13) wrong based on 493 sessions
HideShow timer Statistics
A sequence, a1=64, a2=66, a3=67, an=8+a(n3), which of the following is in the sequence? A. 105 B. 786 C. 966 D. 1025
Official Answer and Stats are available only to registered users. Register/ Login.




Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8187
Location: Pune, India

Re: Sequence question
[#permalink]
Show Tags
19 Jan 2011, 20:30
A sequence, a1=64, a2=66, a3=67, an=8+an3, 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.
_________________
Karishma Veritas Prep GMAT Instructor
Save up to $1,000 on GMAT prep through 8/20! Learn more here >
GMAT selfstudy 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
bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an3, 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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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 DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




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+an3, 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

an=8+an3
Is this an=8+(an)3 OR an=8+a(n3)???



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+an3, 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+an3" so: 64+83 = 69 66+83 = 71 67+83 = 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+an3, 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+an3" so: 64+83 = 69 66+83 = 71 67+83 = 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 (n3) 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+an3, 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
deepaksharma1986 wrote: Bunuel wrote: bhanuvemula wrote: A sequence, a1=64, a2=66, a3=67, an=8+an3, 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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
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 DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



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+an3, 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+an3, which of the
[#permalink]
Show Tags
09 Jul 2013, 09:31



Intern
Joined: 28 May 2012
Posts: 26
Concentration: Finance, General Management
GPA: 3.28
WE: Analyst (Investment Banking)

Re: A sequence, a1=64, a2=66, a3=67, an=8+an3, which of the
[#permalink]
Show Tags
09 Jul 2013, 19:37
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, x2 or x3 is divisible by 8, it would be the answer.



Intern
Joined: 21 May 2013
Posts: 28
Location: India
Concentration: Finance, Marketing

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+an3, 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+an3, 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+an3, 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 >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Manager
Joined: 25 Oct 2013
Posts: 158

Re: A sequence, a1=64, a2=66, a3=67, an=8+a(n3), which of the
[#permalink]
Show Tags
13 Jan 2014, 05:48
Given \(a_n=8+a_{(n3)}\) => \(a_4 = 8+a_1 = 72\) \(a_5 = 8+a_2 = 74\) \(a_6 = 8+a_3 = 75\) 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
_________________
Click on Kudos if you liked the post!
Practice makes Perfect.



SVP
Joined: 06 Sep 2013
Posts: 1851
Concentration: Finance

Re: Sequence question
[#permalink]
Show Tags
06 Feb 2014, 11:12
VeritasPrepKarishma wrote: jlgdr wrote: VeritasPrepKarishma wrote: A sequence, a1=64, a2=66, a3=67, an=8+an3, 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. 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



Intern
Joined: 13 Dec 2013
Posts: 38

Re: A sequence, a1=64, a2=66, a3=67, an=8+a(n3), which of the
[#permalink]
Show Tags
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(n3), which of the &nbs
[#permalink]
15 Jul 2014, 13:00



Go to page
1 2
Next
[ 23 posts ]



