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

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Re: Sequence question [#permalink]
bhanuvemula
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]
an=8+an-3

Is this an=8+(an)-3 OR an=8+a(n-3)???
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Re: Sequence question [#permalink]
techjanson
bhanuvemula
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
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Re: Sequence question [#permalink]
azule45
techjanson
bhanuvemula
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.
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Re: Sequence question [#permalink]
Thanks, Bunuel. It helps.
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Re: Sequence question [#permalink]
Thanks Karishma for the explanation.
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Re: Sequence question [#permalink]
Bunuel
bhanuvemula
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.
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Re: Sequence question [#permalink]
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Expert Reply
deepaksharma1986
Bunuel
bhanuvemula
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.
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Re: Sequence question [#permalink]
deepaksharma1986
Bunuel
bhanuvemula
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.
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Re: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the [#permalink]
Expert Reply
Bumping for review and further discussion.
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Re: A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the [#permalink]
2
Kudos
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]
Bunuel
bhanuvemula
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..
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Re: Sequence question [#permalink]
VeritasPrepKarishma
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
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Re: Sequence question [#permalink]
Expert Reply
jlgdr
VeritasPrepKarishma
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]
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]
VeritasPrepKarishma
jlgdr
VeritasPrepKarishma
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
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Re: A sequence, a1=64, a2=66, a3=67, an=8+a(n-3), which of the [#permalink]
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 [#permalink]
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