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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]

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08 Mar 2012, 22:14
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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]

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09 Mar 2012, 00:29
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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)
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 13 Oct 2011 Posts: 14 Re: The sequence a(n) is defined so that, for all n is greater [#permalink] ### Show Tags 09 Mar 2012, 15:05 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 Intern Joined: 03 Sep 2010 Posts: 16 Re: The Sequence a(n) is... [#permalink] ### Show Tags 10 Mar 2012, 01:38 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 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8026 Location: Pune, India Re: The sequence a(n) is defined so that, for all n is greater [#permalink] ### Show Tags 11 Mar 2012, 02:17 1 This post received KUDOS Expert's post 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 _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Joined: 16 Oct 2010
Posts: 8026
Location: Pune, India
Re: The Sequence a(n) is... [#permalink]

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11 Mar 2012, 02:21
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.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 01 Jun 2012 Posts: 20 Location: United States Concentration: Nonprofit GMAT 1: 720 Q48 V43 GPA: 3.83 Re: The Sequence a(n) is... [#permalink] ### Show Tags 04 Nov 2012, 15:53 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? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8026 Location: Pune, India Re: The Sequence a(n) is... [#permalink] ### Show Tags 04 Nov 2012, 21:29 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Intern
Joined: 20 May 2014
Posts: 33
Re: The sequence a(n) is defined so that, for all n is greater [#permalink]

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03 Jul 2014, 09:41
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
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8026
Location: Pune, India
Re: The sequence a(n) is defined so that, for all n is greater [#permalink]

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03 Jul 2014, 19:38
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...
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 01 Nov 2013 Posts: 318 GMAT 1: 690 Q45 V39 WE: General Management (Energy and Utilities) Re: The sequence a(n) is defined so that, for all n is greater [#permalink] ### Show Tags 16 Mar 2015, 02:02 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 $$a_{n-1}$$. (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 I solved the first option to get a hang of what kind of series comes out... Quickly figured out that only in D it is possible $$a_{n-2} +1$$ is always greater than $$a_{n-1}$$ bcoz both entities are positive and $$a_2=1.5$$ > $$a_1=1$$ _________________ Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time. I hated every minute of training, but I said, 'Don't quit. Suffer now and live the rest of your life as a champion.-Mohammad Ali Director Joined: 07 Aug 2011 Posts: 568 Concentration: International Business, Technology GMAT 1: 630 Q49 V27 Re: The sequence a(n) is defined so that, for all n is greater [#permalink] ### Show Tags 21 Mar 2015, 08:22 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) I have noticed that manahattan gmat questions are very time intensive. i would rather build the series for each of the given options A-E , atleast 2 elements . _________________ Thanks, Lucky _______________________________________________________ Kindly press the to appreciate my post !! Senior Manager Status: love the club... Joined: 24 Mar 2015 Posts: 276 Re: The sequence a(n) is defined so that, for all n is greater [#permalink] ### Show Tags 22 Mar 2018, 02:58 VeritasPrepKarishma wrote: 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. hi mam could you please tell me how to solve every problem without using a pen as egiles mentioned you said in your blog ? thanks in advance, mam Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8026 Location: Pune, India Re: The sequence a(n) is defined so that, for all n is greater [#permalink] ### Show Tags 22 Mar 2018, 05:42 gmatcracker2018 wrote: VeritasPrepKarishma wrote: 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. hi mam could you please tell me how to solve every problem without using a pen as egiles mentioned you said in your blog ? thanks in advance, mam Most actual GMAT questions can be solved using logic and reasoning and do not require equations, too many calculations etc. Usually when I answer questions on GMAT Club, I sit without pen and paper and get the answer orally. After that I start writing the explanation. Very rarely do I need to get up and get my notepad. Though it is a skill one learns through practice. There are things that help such as using ratios, knowing multiplication tables etc. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: The sequence a(n) is defined so that, for all n is greater   [#permalink] 22 Mar 2018, 05:42
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