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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] New post 29 Mar 2007, 17:53
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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
[Reveal] Spoiler: OA
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Re: Sequence question [#permalink] New post 29 Mar 2007, 19:07
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.
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 [#permalink] New post 01 Apr 2007, 20:54
an=8+an-3

Is this an=8+(an)-3 OR an=8+a(n-3)???
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Re: Sequence question [#permalink] New post 27 Sep 2009, 12:48
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.
:thanks 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
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Re: Sequence question [#permalink] New post 15 Oct 2009, 08:13
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.
:thanks 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.
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Re: Sequence question [#permalink] New post 19 Jan 2011, 19:30
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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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Re: Sequence question [#permalink] New post 21 Jan 2011, 12:18
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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


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.
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Re: Sequence question [#permalink] New post 24 Jan 2011, 00:54
Thanks, Bunuel. It helps.
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Re: Sequence question [#permalink] New post 25 Jan 2011, 19:46
Thanks Karishma for the explanation.
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Re: Sequence question [#permalink] New post 07 Mar 2011, 03:32
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.
:thanks 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.
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Re: Sequence question [#permalink] New post 07 Mar 2011, 05:28
Expert's post
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.
:thanks 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.
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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

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Re: Sequence question [#permalink] New post 07 Mar 2011, 06:39
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.
:thanks 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.
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Re: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the [#permalink] New post 09 Jul 2013, 08:31
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Re: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the [#permalink] New post 09 Jul 2013, 18:37
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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.
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Re: Sequence question [#permalink] New post 09 Aug 2013, 04:48
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.
:thanks 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..
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Re: Sequence question [#permalink] New post 18 Dec 2013, 14:38
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!
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Re: Sequence question [#permalink] New post 18 Dec 2013, 18:59
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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.
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Re: A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the [#permalink] New post 13 Jan 2014, 04:48
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] New post 06 Feb 2014, 10:12
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

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
Re: Sequence question   [#permalink] 06 Feb 2014, 10:12
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A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the

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