GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 Oct 2018, 23:37

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

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

A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2

Author Message
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 50007
A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

15 Jun 2016, 01:46
11
57
00:00

Difficulty:

75% (hard)

Question Stats:

61% (01:37) correct 39% (01:48) wrong based on 1537 sessions

HideShow timer Statistics

A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

_________________
Manager
Joined: 25 Jun 2016
Posts: 61
GMAT 1: 780 Q51 V46
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

06 Aug 2016, 06:03
21
2

OG 17 q201 Method 1:

OG 17 q201 Method 2:

General Discussion
Manager
Joined: 04 Apr 2015
Posts: 107
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

15 Jun 2016, 03:00
1
IMO D

An=t so that means product from A1 to An-1 is t. therefore An+1 = A1x.....An-1 x An=txt= t^2
proceeding in same way An+2 will be t^4
SC Moderator
Joined: 13 Apr 2015
Posts: 1693
Location: India
Concentration: Strategy, General Management
GMAT 1: 200 Q1 V1
GPA: 4
WE: Analyst (Retail)
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

15 Jun 2016, 03:10
4
3
An = (A1)(A2)......(An-1) = t
An+1 = t * An = t^2
An+2 = t * An * An+1 = t * t * t^2 = t^4

Intern
Joined: 29 Mar 2016
Posts: 5
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

15 Jun 2016, 03:27
8
10
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, a_2 = 5, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

A) 4t
B) t^2
C) t^3
D) t^4
E) t^8

Given $$a_n = t$$
This means $$a_1 * a_2 * a_3* ..... a_{n-1} = t$$

Therefore $$a_{n+1} = (a_1 * a_2 * a_3* ..... a_{n-1}) * (a_n) = t*t = t^2$$

and $$a_{n+2} = (a_1 * a_2 * a_3* ..... a_{n-1}) * (a_n) * (a_{n+1}) = t*t*t^2 = t^4$$

Answer = $$D$$
Senior Manager
Joined: 18 Jan 2010
Posts: 252
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

15 Jun 2016, 03:49
2
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, a_2 = 5, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

A) 4t
B) t^2
C) t^3
D) t^4
E) t^8

$$a_n = (a_1)(a_2)........(a_{n-1})$$ = t

$$a_{n+1}$$ = $$(a_1)(a_2)........(a_n)$$ = $$(a_1)(a_2)........(a_{n-1})$$*$$a_n$$ = t *t = $$t^2$$

$$a_{n+2}$$ = $$(a_1)(a_2)........(a_{n+1})$$ = $$(a_1)(a_2)........(a_{n-1})$$ * $$a_n$$ *$$a_{n+1}$$

= t * t * $$t^2$$ = $$t^4$$

Manager
Joined: 04 Jun 2015
Posts: 81
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

16 Jul 2017, 08:24
5
3
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, a_2 = 5, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

A) 4t
B) t^2
C) t^3
D) t^4
E) t^8

Say n = 3 [given n>2]. Hence we must find value of a5

a1 = 3,
a2 = 5,
a3 = 5*3 = 15 = t [an = t given]
a4 = 15*15
a5= 15*15*15*15 = t^4
_________________

Sortem sternit fortem!

Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1314
Location: Malaysia
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

29 Aug 2017, 22:32
1
1
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

$$a_3 = a_n = a_1*a_2 = t$$
$$a_4 = a_{n + 1} = a_1*a_2*a_3 = t*a_3 = t*t = t^2$$
$$a_5 = a_{n + 2} = a_1*a_2*a_3*a_4 = t*t*(t^2) = t^4$$
_________________

"Be challenged at EVERY MOMENT."

“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”

"Each stage of the journey is crucial to attaining new heights of knowledge."

Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2830
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

05 Sep 2017, 18:06
1
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

We are given a sequence in which every term in the sequence after a(2) is the product of all terms in the sequence preceding it. So:

a(n+1) = a(n) x a(n-1) x ... x a(2) x a(1)

By the same reasoning, we have:

a(n) = a(n-1) x a(n-2) x ... x a(2) x a(1)

We can substitute a(n-1) x... x a(2) x a(1) in the a(n+1) equation for a(n), so we have a(n+1) = a(n) x a(n).

However, recall that a(n) = t, so a(n+1) = t x t = t^2. By the same reasoning, we have:

a(n+2) = a(n+1) x a(n) x a(n-1) x ... x a(2) x a(1)

However, a(n) x a(n-1) x .... x a(2) x a(1) = a(n+1) and a(n+1) = t^2, so:

a(n+2) = a(n+1) x a(n+1) = t^2 x t^2 = t^4

_________________

Jeffery Miller

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Intern
Joined: 13 Sep 2012
Posts: 9
Location: India
GMAT 1: 680 Q48 V35
GPA: 3.6
A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

13 Oct 2017, 19:28
1
1
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

Observe/Create a sequence:
A3 = A1*A2
A4 = A3*A1*A2 = A3*A3
A5 = A4*A4
.
.
.
An = t
An+1 = An*An = t*t = t^2
An+2 = An+1*An+1 = t^2*t^2 = t^4

D.
CEO
Joined: 12 Sep 2015
Posts: 3021
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

15 Nov 2017, 13:49
3
Top Contributor
1
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

Let's list a few terms....
term1 = 3
term2 = 5
term3 = (term2)(term1) = (5)(3) = 15 (term2)(term1)
term4 = (term3)(term2)(term1) = (15)(5)(3) = 15²
term5 = (term4)(term3)(term2)(term1) = (15²)(15)(5)(3) = 15⁴
term6 = (term5)(term4)(term3)(term2)(term1) = (15⁴)(15²)(15)(5)(3) = 15⁸

At this point, we can see the pattern.

Continuing, we get....
term7 = 15^16
term8 = 15^32

Each term in the sequence is equal to the SQUARE of term before it

If term_n =t and n > 2, what is the value of term_n+2 in terms of t?
So, term_n = t
term_n+1 = t²
term_n+2 = t⁴

Cheers,
Brent
_________________

Brent Hanneson – GMATPrepNow.com

Director
Joined: 09 Mar 2016
Posts: 942
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

02 Jan 2018, 05:58
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

Bunuel - can I call the above mentioned problem as a recursive sequence ? I know this formula An= n1+(n-1)*d but could not apply it here.
Director
Joined: 09 Mar 2016
Posts: 942
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

02 Jan 2018, 06:00
hazelnut wrote:
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

$$a_3 = a_n = a_1*a_2 = t$$
$$a_4 = a_{n + 1} = a_1*a_2*a_3 = t*a_3 = t*t = t^2$$
$$a_5 = a_{n + 2} = a_1*a_2*a_3*a_4 = t*t*(t^2) = t^4$$

Hello hazelnut - how did you figure out that A_n+2 is the fifth term and not sixth ?
Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1314
Location: Malaysia
A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

02 Jan 2018, 06:39
dave13 wrote:
hazelnut wrote:
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

$$a_3 = a_n = a_1*a_2 = t$$
$$a_4 = a_{n + 1} = a_1*a_2*a_3 = t*a_3 = t*t = t^2$$
$$a_5 = a_{n + 2} = a_1*a_2*a_3*a_4 = t*t*(t^2) = t^4$$

Hello hazelnut - how did you figure out that A_n+2 is the fifth term and not sixth ?

let n = 3. $$a_3 = a_n = a_1*a_2 = t$$
_________________

"Be challenged at EVERY MOMENT."

“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”

"Each stage of the journey is crucial to attaining new heights of knowledge."

Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 267
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

20 Mar 2018, 22:33
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, $$a_2 = 5$$, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

(A) 4t
(B) t^2
(C) t^3
(D) t^4
(E) t^8

an = t = .......(an - 1)

(an + 1) = .......(an - 1) * an = t^2

(an + 2) = .....(an - 1) * an * (an + 1)

=) t^2 * t ^2 = t^4 = D the answer

thanks
Director
Joined: 09 Mar 2016
Posts: 942
A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

05 May 2018, 03:14
Sash143 wrote:
Bunuel wrote:
A sequence of numbers $$a_1$$, $$a_2$$, $$a_3$$,…. is defined as follows: $$a_1 = 3$$, a_2 = 5, and every term in the sequence after $$a_2$$ is the product of all terms in the sequence preceding it, e.g, $$a_3 = (a_1)(a_2)$$ and $$a4 = (a_1)(a_2)(a_3)$$. If $$a_n =t$$ and $$n > 2$$, what is the value of $$a_{n+2}$$ in terms of t?

A) 4t
B) t^2
C) t^3
D) t^4
E) t^8

Say n = 3 [given n>2]. Hence we must find value of a5

a1 = 3,
a2 = 5,
a3 = 5*3 = 15 = t [an = t given]
a4 = 15*15
a5= 15*15*15*15 = t^4

Hello pushpitkc, hope you are having an awesome gmat weekend

i have a few questons regarding the above solution, so let me break it down in into following clauses

1.) it says "Say n = 3 [given n>2]. Hence we must find value of a5" my question what does $$n$$ mean is it the last term ? based on which rule do we concude " If n = 3 [given n>2]. Hence we must find value of $$a_5$$ i mean why if n = 3 then we need to find $$a_5$$ and not for example $$a_6$$ ?

2.) if $$a_3 = a_1*a_2 = 3*5 = 15$$, then to find $$a_4$$ we need to perform following $$a_1*a_2*a_3 = 3*5*15 = 225$$

and following this logic in order to find $$a_5$$we need to do this $$a_1*a_2*a_3*a_4 = 3*5*15*225 = 3375$$

i cant understand why in the solution above to find $$a_5$$ we multiply by the same numbers $$a_5= 15*15*15*15 = t^4$$, to find the next term shoudlnt we mupltiply by all previous terms

3.) Also based on this solution $$a_5= 15*15*15*15 = t^4$$ why the correct answer is $$t^4$$ because there are FOUR numbers 15 ? but there could be infinite NUMBERS of 15

And the last question is it geometric sequence question ?

many thanks for taking time to explain

enjoy the weekend
Intern
Joined: 15 Sep 2018
Posts: 23
A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

03 Oct 2018, 04:53
1
For problems like this, it’s best that we observe the first few terms of the sequence to see if there’s any noticeable pattern among the results.

Since it’s given that $$a_1=3$$ and $$a_2=5$$, this means that

$$a_3=(a_1 )(a_2 )=3(5)=15=15^1$$
$$a_4=(a_1 )(a_2 )(a_3 )=(3)(5)(15)=225=15^2$$
$$a_5=(a_1 )(a_2 )(a_3 )(a_4 )=(3)(5)(15)(15^2 )= 15^4 =(15^2 )^2$$
$$a_6=(3)(5)(15)(15^2 )(15^4 )= 15^8=(15^4 )^2$$

As can be seen, as the sequence progresses, the next term would be the square of the term before it.

So, if we start at $$a_n=t$$, the next term would be $$t^2$$ and term after that would be $$(t^2 )^2=t^4$$.

$$a_n=t$$
$$a_{(n+1)}=t^2$$
$$a_{(n+2 )}=(t^2 )^2=t^4$$

Hence, the final answer is .
Director
Joined: 09 Mar 2016
Posts: 942
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2  [#permalink]

Show Tags

03 Oct 2018, 05:14
EMPOWERMathExpert wrote:
For problems like this, it’s best that we observe the first few terms of the sequence to see if there’s any noticeable pattern among the results.

Since it’s given that $$a_1=3$$ and $$a_2=5$$, this means that

$$a_3=(a_1 )(a_2 )=3(5)=15=15^1$$
$$a_4=(a_1 )(a_2 )(a_3 )=(3)(5)(15)=225=15^2$$
$$a_5=(a_1 )(a_2 )(a_3 )(a_4 )=(3)(5)(15)(15^2 )= 15^4 =(15^2 )^2$$
$$a_6=(3)(5)(15)(15^2 )(15^4 )= 15^8=(15^4 )^2$$

As can be seen, as the sequence progresses, the next term would be the square of the term before it.

So, it we start at $$a_n=t$$, the next term would be $$t^2$$ and term after that would be $$(t^2 )^2=t^4$$.

$$a_n=t$$
$$a_{(n+1)}=t^2$$
$$a_{(n+2 )}=(t^2 )^2=t^4$$

Hence, the final answer is .

Hey EMPOWERMathExpert i have been waiting for your explanation for many years:-) what took you so long welcome to Club of forever students
Re: A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 &nbs [#permalink] 03 Oct 2018, 05:14
Display posts from previous: Sort by