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

It is currently 20 Sep 2018, 15:24

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.

Close

Request Expert Reply

Confirm Cancel

A certain sequence starts with term_1

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

CEO
CEO
User avatar
D
Joined: 12 Sep 2015
Posts: 2865
Location: Canada
A certain sequence starts with term_1  [#permalink]

Show Tags

New post 06 Apr 2017, 08:07
3
Top Contributor
7
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

58% (02:45) correct 42% (02:53) wrong based on 137 sessions

HideShow timer Statistics

A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80

* Kudos for all correct solutions

_________________

Brent Hanneson – GMATPrepNow.com
Image
Sign up for our free Question of the Day emails

Most Helpful Community Reply
Senior Manager
Senior Manager
User avatar
G
Joined: 19 Apr 2016
Posts: 274
Location: India
GMAT 1: 570 Q48 V22
GMAT 2: 640 Q49 V28
GPA: 3.5
WE: Web Development (Computer Software)
A certain sequence starts with term_1  [#permalink]

Show Tags

New post Updated on: 06 Apr 2017, 08:55
7
2
GMATPrepNow wrote:
A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80

* Kudos for all correct solutions


term_n = 16^(2n - 1) = \(2^{8n-4}\)

term_1 = \(2^{8-4} = 2^4\)

term_2 = \(2^{16-4} = 2^{12}\)

term_3 = \(2^{24-4} = 2^{20}\)
.
.
.
term_k = \(2^{8k-4}\)

product of the first k terms of the sequence is 2^1600

term_1 * term_2 * term_3 ......* term_k =\(2^{1600}\)

\(2^4*2^{12}*2^{20}.....*2^{8k-4}=2^{1600}\)

\(2^{4+12+20+....+8k-4}=2^{1600}\)

4+12+20+...8k-4 = 1600

sum of the above AP = (k/2)(4+8k-4) = \(4K^2\)

\(4K^2=1600\)

\(K^2=400\)

k=20

Hence option C is correct
Hit Kudos if you liked it 8-)

Originally posted by 0akshay0 on 06 Apr 2017, 08:54.
Last edited by 0akshay0 on 06 Apr 2017, 08:55, edited 1 time in total.
General Discussion
Senior Manager
Senior Manager
avatar
G
Joined: 24 Apr 2016
Posts: 333
Re: A certain sequence starts with term_1  [#permalink]

Show Tags

New post 06 Apr 2017, 08:53
1
1
term_1 = 16 ^ (2-1) = 16 ^ 1
term_2 = 16 ^ (4-1) = 16 ^ 3
term_3 = 16 ^ (6-1) = 16 ^ 5

So we see a pattern forming.

question says that (16 ^ 1) * (16 ^ 3) * (16 ^ 5) *.... [16 ^ (2k-1)] = 2^1600 = 16 ^ 400

hence the powers of 16 on the left side add up to 400

1+3+5 +....2k-1 = 400

Using the formula :: n/2 [2+(n-1)2] = 400

n/2[2+2n-2] = 400 ==> n/2[2n] = 400 ==> n^2 = 20^2 ==> n=20

Answer is C) 20
Intern
Intern
avatar
B
Joined: 20 Feb 2017
Posts: 15
Location: India
Concentration: Operations, International Business
GMAT 1: 560 Q47 V21
GPA: 3.05
WE: Other (Entertainment and Sports)
GMAT ToolKit User
A certain sequence starts with term_1  [#permalink]

Show Tags

New post 06 Apr 2017, 08:55
3
Since,

nth term = term_n = 16^(2n - 1), the series will be 16,16^3,16^5,16^7,.....

Product of 2 terms = 16^4
Product of 3 terms = 16^9
Product of 4 terms = 16^16
Product of n terms = 16^n^2

2^1600 = 16^400
ie; n^2 = 400, so n or here k = 20
CEO
CEO
User avatar
D
Joined: 12 Sep 2015
Posts: 2865
Location: Canada
Re: A certain sequence starts with term_1  [#permalink]

Show Tags

New post 07 Apr 2017, 07:34
3
Top Contributor
1
GMATPrepNow wrote:
A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80


First notice that 2n - 1, will be ODD for all integer values of n. For example:
If n = 1, then 2n - 1 = 2(1) - 1 = 1
If n = 2, then 2n - 1 = 2(2) - 1 = 3
If n = 3, then 2n - 1 = 2(3) - 1 = 5
If n = 4, then 2n - 1 = 2(4) - 1 = 7
.
.
.
etc.


Now notice what happens when we add consecutive ODD numbers (starting with 1)
The first 1 ODD number: 1 = 1 (and 1 = 1²)
The first 2 ODD numbers: 1 + 3 = 4 (and 4 = 2²)
The first 3 ODD numbers: 1 + 3 + 5 = 9 (and 9 = 3²)
The first 4 ODD numbers: 1 + 3 + 5 + 7 = 16 (and 16 = 4²)
The first 5 ODD numbers: 1 + 3 + 5 + 7 + 9 = 25 (and 25 = 5²)
.
.
.
In general, the sum of the first k ODD numbers = k²

Now onto the question!!!

term_n = 16^(2n - 1)
term_1 = 16^(2(1) - 1) = 16^1
term_2 = 16^(2(2) - 1) = 16^3
term_3 = 16^(3(3) - 1) = 16^5
term_4 = 16^(2(4) - 1) = 16^7
etc

So, the PRODUCT of the first k terms = (16^1)(16^3)(16^5)(16^7)(16^9). . . (16^??)
When we multiply powers with the same base, we ADD the exponents.
So, the PRODUCT of the first k terms = 16^(1 + 3 + 5 + 7 + . . . ??)

Notice that the exponent here is equal to the SUM of the first k ODD numbers.
Well, we already know that the sum of the first k ODD numbers = k²
So, the PRODUCT of the first k terms = 16^()

We're told that the PRODUCT of the first k terms is 2^1600
So, we can write: 16^() = 2^1600

We need the same base, so let's rewrite 16 as 2^4
We get: (2^4)^() = 2^1600
Apply power of a power law: 2^(4k²) = 2^1600
This means that 4k² = 1600
Divide both sides by 4 to get: k² = 400
Solve: k = 20 or -20
Since -20 makes no sense, we know that k = 20

In other words, the PRODUCT of the first 20 terms of the sequence is 2^1600,

Answer:

Cheers,
Brent
_________________

Brent Hanneson – GMATPrepNow.com
Image
Sign up for our free Question of the Day emails

Manager
Manager
avatar
S
Joined: 13 Dec 2013
Posts: 161
Location: United States (NY)
Concentration: Nonprofit, International Business
GMAT 1: 710 Q46 V41
GMAT 2: 720 Q48 V40
GPA: 4
WE: Consulting (Consulting)
Reviews Badge
Re: A certain sequence starts with term_1  [#permalink]

Show Tags

New post 12 Apr 2017, 17:33
1
0akshay0 wrote:
GMATPrepNow wrote:
A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80

* Kudos for all correct solutions


term_n = 16^(2n - 1) = \(2^{8n-4}\)

term_1 = \(2^{8-4} = 2^4\)

term_2 = \(2^{16-4} = 2^{12}\)

term_3 = \(2^{24-4} = 2^{20}\)
.
.
.
term_k = \(2^{8k-4}\)

product of the first k terms of the sequence is 2^1600

term_1 * term_2 * term_3 ......* term_k =\(2^{1600}\)

\(2^4*2^{12}*2^{20}.....*2^{8k-4}=2^{1600}\)

\(2^{4+12+20+....+8k-4}=2^{1600}\)

4+12+20+...8k-4 = 1600

sum of the above AP = (k/2)(4+8k-4) = \(4K^2\)

\(4K^2=1600\)

\(K^2=400\)

k=20

Hence option C is correct
Hit Kudos if you liked it 8-)


Thanks for the post. Can you go over this part?

sum of the above AP = (k/2)(4+8k-4) = \(4K^2\)
Intern
Intern
avatar
B
Joined: 08 Dec 2016
Posts: 39
CAT Tests
Re: A certain sequence starts with term_1  [#permalink]

Show Tags

New post 10 May 2017, 07:08
1
Cez005 wrote:
0akshay0 wrote:
GMATPrepNow wrote:
A certain sequence starts with term_1

For any term in the sequence, term_n = 16^(2n - 1)

If the PRODUCT of the first k terms of the sequence is 2^1600, what is the value of k?

A) 5
B) 10
C) 20
D) 40
E) 80

* Kudos for all correct solutions


term_n = 16^(2n - 1) = \(2^{8n-4}\)

term_1 = \(2^{8-4} = 2^4\)

term_2 = \(2^{16-4} = 2^{12}\)

term_3 = \(2^{24-4} = 2^{20}\)
.
.
.
term_k = \(2^{8k-4}\)

product of the first k terms of the sequence is 2^1600

term_1 * term_2 * term_3 ......* term_k =\(2^{1600}\)

\(2^4*2^{12}*2^{20}.....*2^{8k-4}=2^{1600}\)

\(2^{4+12+20+....+8k-4}=2^{1600}\)

4+12+20+...8k-4 = 1600

sum of the above AP = (k/2)(4+8k-4) = \(4K^2\)

\(4K^2=1600\)

\(K^2=400\)

k=20

Hence option C is correct
Hit Kudos if you liked it 8-)


Thanks for the post. Can you go over this part?

sum of the above AP = (k/2)(4+8k-4) = \(4K^2\)

-------------------------------------------------------------------------
Formula: #terms(Val_max+Val_min)/2 = sum of equidistant numbers
Intern
Intern
avatar
B
Joined: 04 Jul 2018
Posts: 10
Re: A certain sequence starts with term_1  [#permalink]

Show Tags

New post 21 Aug 2018, 05:46
quantumliner wrote:
term_1 = 16 ^ (2-1) = 16 ^ 1
term_2 = 16 ^ (4-1) = 16 ^ 3
term_3 = 16 ^ (6-1) = 16 ^ 5

So we see a pattern forming.

question says that (16 ^ 1) * (16 ^ 3) * (16 ^ 5) *.... [16 ^ (2k-1)] = 2^1600 = 16 ^ 400

hence the powers of 16 on the left side add up to 400

1+3+5 +....2k-1 = 400

Using the formula :: n/2 [2+(n-1)2] = 400

n/2[2+2n-2] = 400 ==> n/2[2n] = 400 ==> n^2 = 20^2 ==> n=20

Answer is C) 20




How do I get the formula : n/2 [2+(n-1)2] = 400 ???
Is there any previous post that explains this formula?
Re: A certain sequence starts with term_1 &nbs [#permalink] 21 Aug 2018, 05:46
Display posts from previous: Sort by

A certain sequence starts with term_1

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  

Events & Promotions

PREV
NEXT


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

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

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®.