Find all School-related info fast with the new School-Specific MBA Forum

It is currently 30 Aug 2015, 23:36
GMAT Club Tests

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

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

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
2 KUDOS received
Intern
Intern
avatar
Joined: 13 Oct 2011
Posts: 16
Followers: 0

Kudos [?]: 11 [2] , given: 16

GMAT ToolKit User Reviews Badge
The sequence a(n) is defined so that, for all n is greater [#permalink] New post 08 Mar 2012, 21:14
2
This post received
KUDOS
7
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

58% (03:15) correct 42% (01:56) wrong based on 156 sessions
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
Expert Post
5 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 5853
Location: Pune, India
Followers: 1480

Kudos [?]: 7956 [5] , given: 190

Re: The Sequence a(n) is... [#permalink] New post 08 Mar 2012, 23:29
5
This post received
KUDOS
Expert's post
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

1 KUDOS received
Intern
Intern
avatar
Joined: 13 Oct 2011
Posts: 16
Followers: 0

Kudos [?]: 11 [1] , given: 16

GMAT ToolKit User Reviews Badge
Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 09 Mar 2012, 07:56
1
This post received
KUDOS
thank you so much :)
Intern
Intern
avatar
Joined: 13 Oct 2011
Posts: 16
Followers: 0

Kudos [?]: 11 [0], given: 16

GMAT ToolKit User Reviews Badge
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
Intern
Intern
User avatar
Joined: 03 Sep 2010
Posts: 16
Followers: 2

Kudos [?]: 8 [0], given: 14

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
Expert Post
1 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 5853
Location: Pune, India
Followers: 1480

Kudos [?]: 7956 [1] , given: 190

Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 11 Mar 2012, 01:17
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

Veritas Prep Reviews

Expert Post
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 5853
Location: Pune, India
Followers: 1480

Kudos [?]: 7956 [0], given: 190

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.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199

Veritas Prep Reviews

Current Student
avatar
Joined: 01 Jun 2012
Posts: 21
Location: United States
Concentration: Nonprofit
GMAT 1: 720 Q48 V43
GPA: 3.83
Followers: 0

Kudos [?]: 8 [0], given: 15

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?
Expert Post
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 5853
Location: Pune, India
Followers: 1480

Kudos [?]: 7956 [0], given: 190

Re: The Sequence a(n) is... [#permalink] New post 04 Nov 2012, 20:29
Expert's post
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

Veritas Prep Reviews

GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 09 Sep 2013
Posts: 6121
Followers: 342

Kudos [?]: 70 [0], given: 0

Premium Member
Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 21 Feb 2014, 11:35
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Intern
Intern
avatar
Joined: 20 May 2014
Posts: 40
Followers: 0

Kudos [?]: 1 [0], given: 1

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 :)
Expert Post
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 5853
Location: Pune, India
Followers: 1480

Kudos [?]: 7956 [0], given: 190

Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 03 Jul 2014, 18:38
Expert's post
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
Senior Manager
User avatar
Joined: 01 Nov 2013
Posts: 290
WE: General Management (Energy and Utilities)
Followers: 2

Kudos [?]: 75 [0], given: 392

Reviews Badge
Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 16 Mar 2015, 01:02
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
Director
User avatar
Joined: 07 Aug 2011
Posts: 588
Concentration: International Business, Technology
GMAT 1: 630 Q49 V27
Followers: 2

Kudos [?]: 237 [0], given: 75

GMAT ToolKit User
Re: The sequence a(n) is defined so that, for all n is greater [#permalink] New post 21 Mar 2015, 07:22
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 Image to appreciate my post !! :-)

Re: The sequence a(n) is defined so that, for all n is greater   [#permalink] 21 Mar 2015, 07:22
    Similar topics Author Replies Last post
Similar
Topics:
1 Experts publish their posts in the topic The sequence S is defined as follows for all n ≥ 1: The sum of the Bunuel 7 17 Jun 2015, 05:00
5 Experts publish their posts in the topic If sequence T is defined for all positive integers n such that tn +1 = viktorija 3 12 Jan 2015, 14:18
5 Experts publish their posts in the topic In the sequence g_n defined for all positive integer daviesj 3 22 Dec 2012, 06:28
11 Experts publish their posts in the topic For all positive integers n, the sequence An is defined by daviesj 9 22 Dec 2012, 05:34
11 Experts publish their posts in the topic The sequence f(n) = (2n)! ÷ n! is defined for all positive enigma123 2 15 Jan 2012, 14:28
Display posts from previous: Sort by

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

  Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Privacy Policy| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.