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The sequence a(n) is defined so that, for all n is greater

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The sequence a(n) is defined so that, for all n is greater [#permalink] New post 08 Mar 2012, 21:14
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The sequence a_n is defined so that, for all n is greater than or equal to 3, a_n is the greater of a_{n-2} +1 and a_{n-1}. (If the two quantities are the same, then a_n is equal to either of them.) Which of the following values of a_1 and a_2 will produce a sequence in which no value is repeated?

A. a_1=-1, a_2=-1.5
B. a_1=-1, a_2=1
C. a_1=1, a_2=-1
D. a_1=1, a_2=1.5
E. a_1=1.5, a_2=1

I am having trouble understanding what the question is asking and also solving it
[Reveal] Spoiler: OA
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Re: The Sequence a(n) is... [#permalink] New post 08 Mar 2012, 23:29
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shawndx wrote:
The sequence a(n) is defined so that, for all n is greater than or equal to 3, a(n) is the greater of (a(n-2) +1) and (an-1). (If the two quantities are the same, then an is equal to either of them.) Which of the following values of a1 and a2 will produce a sequence in which no value is repeated?

a) a1=-1, a2=-1.5
b) a1=-1, a2=1
c) a1=1, a2=-1
d) a1=1, a2=1.5
e) a1=1.5, a2=1

I am having trouble understanding what the question is asking and also solving it


Take one line of the question at a time and try to make sense of it. Sequence questions seem daunting due to all the subscripts but they are pretty straight forward, generally.

Given: a_n = Greater of (a_{n-2} + 1, a_{n-1}) (n is 3 or greater)
This means that starting from the third term, every term is the greater of (one more than previous to previous term, the previous term)

If we want that every term in the sequence should be unique, a_n should not be equal to the previous term. It should be equal to 'one more than previous to previous term'

So what can you deduce about a_1 and a_2?
1. a_1 + 1 should be greater than a_2 so that a_3 \neq a_2. Reject option (B)

2. To ensure that a_4 \neq a_3, a_4 = a_2 + 1. Therefore, a_2 + 1 > a_3

Option (A) a_3 = 0 which is greater than a_2 + 1 (= -0.5) so reject it.
Option (C) a_3 = 2 which is greater than a_2 + 1 (= 0) so reject it.
Option (E) a_3 = 2.5 which is greater than a_2 + 1 (= 2) so reject it.

Answer must be option (D). a_3 = 2 which is less than a_2 + 1.

OR if you want to think the logical way, realize that the first term must be smaller than the second term but the difference between them should be less than 1 (so that when 1 is added, it becomes more than the second term). If you understand this, then you can quickly jump to option (D)
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Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 09 Mar 2012, 07:56
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thank you so much :)
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Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 09 Mar 2012, 14:05
I do not understand the rule though... so if we were to just substitute values, how would one do that... if n were 3 or 4
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Re: The Sequence a(n) is... [#permalink] New post 10 Mar 2012, 00:38
VeritasPrepKarishma wrote:
shawndx wrote:
The sequence a(n) is defined so that, for all n is greater than or equal to 3, a(n) is the greater of (a(n-2) +1) and (an-1). (If the two quantities are the same, then an is equal to either of them.) Which of the following values of a1 and a2 will produce a sequence in which no value is repeated?

a) a1=-1, a2=-1.5
b) a1=-1, a2=1
c) a1=1, a2=-1
d) a1=1, a2=1.5
e) a1=1.5, a2=1

I am having trouble understanding what the question is asking and also solving it


Take one line of the question at a time and try to make sense of it. Sequence questions seem daunting due to all the subscripts but they are pretty straight forward, generally.

Given: a_n = Greater of (a_{n-2} + 1, a_{n-1}) (n is 3 or greater)
This means that starting from the third term, every term is the greater of (one more than previous to previous term, the previous term)

If we want that every term in the sequence should be unique, a_n should not be equal to the previous term. It should be equal to 'one more than previous to previous term'

So what can you deduce about a_1 and a_2?
1. a_1 + 1 should be greater than a_2 so that a_3 \neq a_2. Reject option (B)

2. To ensure that a_4 \neq a_3, a_4 = a_2 + 1. Therefore, a_2 + 1 > a_3

Option (A) a_3 = 0 which is greater than a_2 + 1 (= -0.5) so reject it.
Option (C) a_3 = 2 which is greater than a_2 + 1 (= 0) so reject it.
Option (E) a_3 = 2.5 which is greater than a_2 + 1 (= 2) so reject it.

Answer must be option (D). a_3 = 2 which is less than a_2 + 1.

OR if you want to think the logical way, realize that the first term must be smaller than the second term but the difference between them should be less than 1 (so that when 1 is added, it becomes more than the second term). If you understand this, then you can quickly jump to option (D)


@karishma
To be honest what is probability of such questions landing up on your gmat and on what level ...
I think, I would just flip the computer table and walk out rather than solve such types
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Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 11 Mar 2012, 01:17
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shawndx wrote:
I do not understand the rule though... so if we were to just substitute values, how would one do that... if n were 3 or 4


a_3 depends on a_1 and a_2. Given the values of a_1 and a_2, you can find the values of all other terms.

Say if a_1 = 1 and a_2 = 1.5, then
a_3 = Greater (1 + 1, 1.5) = 2
a_4 = Greater (1.5+ 1, 2) = 2.5 etc

On the other hand, if a_1 = 1
a_2 = 1.8, then
a_3 = Greater (1 + 1, 1.8) = 2
a_4 = Greater (1.8+ 1, 2) = 2.8 etc
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Re: The Sequence a(n) is... [#permalink] New post 11 Mar 2012, 01:21
Expert's post
utkarshlavania wrote:

@karishma
To be honest what is probability of such questions landing up on your gmat and on what level ...
I think, I would just flip the computer table and walk out rather than solve such types


The question is not tough Utkarsh. I would suggest you to go one step at a time and try to figure it out. Such questions can be a part of actual GMAT and it's just a 700+, not exceptionally over the top. It looks complicated but most of the steps would be kind of intuitive after some practice. GMAT excels at testing simple concepts wrapped in a twisted package.
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Re: The Sequence a(n) is... [#permalink] New post 04 Nov 2012, 14:53
VeritasPrepKarishma wrote:
shawndx wrote:
The sequence a(n) is defined so that, for all n is greater than or equal to 3, a(n) is the greater of (a(n-2) +1) and (an-1). (If the two quantities are the same, then an is equal to either of them.) Which of the following values of a1 and a2 will produce a sequence in which no value is repeated?

a) a1=-1, a2=-1.5
b) a1=-1, a2=1
c) a1=1, a2=-1
d) a1=1, a2=1.5
e) a1=1.5, a2=1

I am having trouble understanding what the question is asking and also solving it


Take one line of the question at a time and try to make sense of it. Sequence questions seem daunting due to all the subscripts but they are pretty straight forward, generally.

Given: a_n = Greater of (a_{n-2} + 1, a_{n-1}) (n is 3 or greater)
This means that starting from the third term, every term is the greater of (one more than previous to previous term, the previous term)

If we want that every term in the sequence should be unique, a_n should not be equal to the previous term. It should be equal to 'one more than previous to previous term'

So what can you deduce about a_1 and a_2?
1. a_1 + 1 should be greater than a_2 so that a_3 \neq a_2. Reject option (B)

2. To ensure that a_4 \neq a_3, a_4 = a_2 + 1. Therefore, a_2 + 1 > a_3

Option (A) a_3 = 0 which is greater than a_2 + 1 (= -0.5) so reject it.
Option (C) a_3 = 2 which is greater than a_2 + 1 (= 0) so reject it.
Option (E) a_3 = 2.5 which is greater than a_2 + 1 (= 2) so reject it.

Answer must be option (D). a_3 = 2 which is less than a_2 + 1.

OR if you want to think the logical way, realize that the first term must be smaller than the second term but the difference between them should be less than 1 (so that when 1 is added, it becomes more than the second term). If you understand this, then you can quickly jump to option (D)



Hi Karishma,

Thanks for your blog...I have been trying to solve every problem without using a pen per your advice and I am getting better at the quant section as a result.

Here is where I am confused about the above problem:

"If we want that every term in the sequence should be unique, n should not be equal to the previous term. It should be equal to 'one more than previous to previous term'"

I'm reading the rule as A(n) is greater than A(n-2) + 1. Above, you say they must be equal (at least that is how I'm reading it). Overall, the wording of this problem is awkward to me. I've never seen the phrase "...is the greater of". That just means it is great than, right?
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Re: The Sequence a(n) is... [#permalink] New post 04 Nov 2012, 20:29
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egiles wrote:
"If we want that every term in the sequence should be unique, n should not be equal to the previous term. It should be equal to 'one more than previous to previous term'"

I'm reading the rule as A(n) is greater than A(n-2) + 1. Above, you say they must be equal (at least that is how I'm reading it). Overall, the wording of this problem is awkward to me. I've never seen the phrase "...is the greater of". That just means it is great than, right?



There are 2 diff things:

1. 'Is greater than'
x is greater than 4 and 5. This means x is a number greater than 5.

2. 'Is greater of'
x is greater of 4 and 5. This means x = 5.
Look at it this way: x = Greater of (4, 5)
First you find that which number is greater out of 4 and 5. x will be equal to that number.

x = Greater of (Last to last term + 1, Last term)
means find which is greater 'Last to last term + 1' or 'Last term'. X will be equal to that.
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Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 21 Feb 2014, 11:35
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Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 03 Jul 2014, 08:41
VeritasPrepKarishma wrote:
shawndx wrote:
The sequence a(n) is defined so that, for all n is greater than or equal to 3, a(n) is the greater of (a(n-2) +1) and (an-1). (If the two quantities are the same, then an is equal to either of them.) Which of the following values of a1 and a2 will produce a sequence in which no value is repeated?

a) a1=-1, a2=-1.5
b) a1=-1, a2=1
c) a1=1, a2=-1
d) a1=1, a2=1.5
e) a1=1.5, a2=1

I am having trouble understanding what the question is asking and also solving it


Take one line of the question at a time and try to make sense of it. Sequence questions seem daunting due to all the subscripts but they are pretty straight forward, generally.

Given: a_n = Greater of (a_{n-2} + 1, a_{n-1}) (n is 3 or greater)
This means that starting from the third term, every term is the greater of (one more than previous to previous term, the previous term)

If we want that every term in the sequence should be unique, a_n should not be equal to the previous term. It should be equal to 'one more than previous to previous term'

So what can you deduce about a_1 and a_2?
1. a_1 + 1 should be greater than a_2 so that a_3 \neq a_2. Reject option (B)

2. To ensure that a_4 \neq a_3, a_4 = a_2 + 1. Therefore, a_2 + 1 > a_3

Option (A) a_3 = 0 which is greater than a_2 + 1 (= -0.5) so reject it.
Option (C) a_3 = 2 which is greater than a_2 + 1 (= 0) so reject it.
Option (E) a_3 = 2.5 which is greater than a_2 + 1 (= 2) so reject it.

Answer must be option (D). a_3 = 2 which is less than a_2 + 1.

OR if you want to think the logical way, realize that the first term must be smaller than the second term but the difference between them should be less than 1 (so that when 1 is added, it becomes more than the second term). If you understand this, then you can quickly jump to option (D)


KARISHMA (SORRY FOR CAPS JUST TRYING TO HIGHLIGHT WHERE MY QUESTION IS):

WHEN YOU SAY N IS 3 OR GREATER TOWARDS THE BEGINNING OF YOUR ANSWER HOW EXACTLY DID YOU DEDUCE THAT?
SECONDLY WHEN YOU ARE TESTING ANSWER CHOICES HOW EXACTLY DO YOU GET THE VALUE OF A3? FOR EXAMPLE WHEN YOU TEST CHOICE C HOW DID YOU GET A3 = 2?

THANKS :)
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Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 03 Jul 2014, 18:38
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sagnik2422 wrote:
KARISHMA (SORRY FOR CAPS JUST TRYING TO HIGHLIGHT WHERE MY QUESTION IS):

WHEN YOU SAY N IS 3 OR GREATER TOWARDS THE BEGINNING OF YOUR ANSWER HOW EXACTLY DID YOU DEDUCE THAT?
SECONDLY WHEN YOU ARE TESTING ANSWER CHOICES HOW EXACTLY DO YOU GET THE VALUE OF A3? FOR EXAMPLE WHEN YOU TEST CHOICE C HOW DID YOU GET A3 = 2?

THANKS :)


Since all previous quotes have a lighter background, it is easy to see where your question is so don't worry about that.

WHEN YOU SAY N IS 3 OR GREATER TOWARDS THE BEGINNING OF YOUR ANSWER HOW EXACTLY DID YOU DEDUCE THAT?

It is given in the question: "The sequence a(n) is defined so that, for all n is greater than or equal to 3".

SECONDLY WHEN YOU ARE TESTING ANSWER CHOICES HOW EXACTLY DO YOU GET THE VALUE OF A3? FOR EXAMPLE WHEN YOU TEST CHOICE C HOW DID YOU GET A3 = 2?

We know that An is greater of A(n-2) + 1 and A(n-1). This means An will be either (A(n-2)+ 1) or A(n-1), whichever is greater!
Now use options:

a) a1=-1, a2=-1.5

A3 = Greater of (A(1)+1, A(2))
A3 = Greater of (-1+1, -1.5)
A3 = Greater of (0, -1.5)
A3 = 0

c) a1=1, a2=-1

A3 = Greater of (A(1)+1, A(2))
A3 = Greater of (1+1, -1)
A3 = Greater of (2, -1)
A3 = 2

and so on...
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Re: The sequence a(n) is defined so that, for all n is greater   [#permalink] 03 Jul 2014, 18:38
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