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

 It is currently 20 Feb 2019, 15:50

### 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

## Events & Promotions

###### Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT Prep Hour

February 20, 2019

February 20, 2019

08:00 PM EST

09:00 PM EST

Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST

February 21, 2019

February 21, 2019

10:00 PM PST

11:00 PM PST

Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

# The sequence S is defined as follows for all n ≥ 1: The sum of the

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 53020
The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

17 Jun 2015, 05:00
00:00

Difficulty:

55% (hard)

Question Stats:

63% (02:09) correct 37% (02:16) wrong based on 130 sessions

### HideShow timer Statistics

The sequence S is defined as follows for all n ≥ 1:

$$S_n=(-1)^n*\frac{1}{n(n+1)}$$

The sum of the first 10 terms of S is:

(A) Between –1 and –1/2
(B) Between –1/2 and 0
(C) Between 0 and 1/2
(D) Between 1/2 and 1
(E) Greater than 1

_________________
CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2788
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

17 Jun 2015, 05:33
1
Bunuel wrote:
The sequence S is defined as follows for all n ≥ 1:

$$S_n=(-1)^n*\frac{1}{n(n+1)}$$

The sum of the first 10 terms of S is:

(A) Between –1 and –1/2
(B) Between –1/2 and 0
(C) Between 0 and 1/2
(D) Between 1/2 and 1
(E) Greater than 1

Given: $$S_n=(-1)^n*\frac{1}{n(n+1)}$$

i.e. $$S_1=(-1)^1*\frac{1}{1(1+1)} = -1/2$$
i.e. $$S_2=(-1)^2*\frac{1}{2(2+1)} = 1/6$$
i.e. $$S_3=(-1)^3*\frac{1}{3(3+1)} = -1/12$$
i.e. $$S_4=(-1)^4*\frac{1}{4(4+1)} = 1/20$$
i.e. $$S_5=(-1)^5*\frac{1}{5(5+1)} = -1/30$$
... and so on

Sum of First 10 terms = (-1/2) + (1/6) + (-1/12) + (1/20) + (-1/30) + (1/42) .... and so on

Method-1

Substitute the approximate values of terms

Sum of First 10 terms = -0.5 + 0.16 - 0.083 + 0.05 - 0.033 + 0.023 - ...

Sum of First 10 terms = (-0.5 + 0.16) + (-0.083 + 0.05) + (-0.033 + 0.023) + ...
Sum of First 10 terms = (-0.34) + (-0.03) + (-0.01) + ... = approx. (-0.4) because the next terms will be very small

i.e. It falls between 0 and -0.5

_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2788
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

17 Jun 2015, 06:01
Bunuel wrote:
The sequence S is defined as follows for all n ≥ 1:

$$S_n=(-1)^n*\frac{1}{n(n+1)}$$

The sum of the first 10 terms of S is:

(A) Between –1 and –1/2
(B) Between –1/2 and 0
(C) Between 0 and 1/2
(D) Between 1/2 and 1
(E) Greater than 1

Given: $$S_n=(-1)^n*\frac{1}{n(n+1)}$$

i.e. $$S_1=(-1)^1*\frac{1}{1(1+1)} = -1/2$$
i.e. $$S_2=(-1)^2*\frac{1}{2(2+1)} = 1/6$$
i.e. $$S_3=(-1)^3*\frac{1}{3(3+1)} = -1/12$$
i.e. $$S_4=(-1)^4*\frac{1}{4(4+1)} = 1/20$$
i.e. $$S_5=(-1)^5*\frac{1}{5(5+1)} = -1/30$$
... and so on

Sum of First 10 terms = (-1/2) + (1/6) + (-1/12) + (1/20) + (-1/30) + (1/42) .... and so on

Method-2

i.e. Sum of First 10 terms = (-1/2) + (1/12) + (1/60) + .... and so on

i.e. Sum of First 10 terms = (-1/2) + some very small positive values

i.e. Sum of First 10 terms = Greater that (-1/2) but less than Zero

i.e. It falls between 0 and -0.5

_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Math Expert
Joined: 02 Aug 2009
Posts: 7334
The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

17 Jun 2015, 06:23
Bunuel wrote:
The sequence S is defined as follows for all n ≥ 1:

$$S_n=(-1)^n*\frac{1}{n(n+1)}$$

The sum of the first 10 terms of S is:

(A) Between –1 and –1/2
(B) Between –1/2 and 0
(C) Between 0 and 1/2
(D) Between 1/2 and 1
(E) Greater than 1

Hi,
i would do this Q in two steps...
1) first step would be to eliminate all wrong answers..
as we can see the first term is negative ,next +ive and so on.. basically all odd numbers are -ive and even numbers are +ive..
Also the denominator is increasing in subsequent term, so the answer has to be a negative number that is <0..
only A and B are left..
2) second step to close on the answer between the two requires a bit of calculations...
$$\frac{1}{(n(n+1))}$$=$$\frac{1}{n}-\frac{1}{(n+1)}$$...
so first two terms give us=$$-1+\frac{1}{2}+\frac{1}{2}-\frac{1}{3}=-\frac{1}{3}$$..
next two terms=$$-\frac{1}{3}+\frac{1}{4}+\frac{1}{4}-\frac{1}{5}=-\frac{1}{30}$$...
we can see the vaue gets significantly smaller with every next two terms so we can estimate that the answer will be slightly lesser than -1/3 but should be more than -1/2..
so ans B.. we can actually next two terms as -1/105.. and can find other two to get to exact answer but may not require that..
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

Manager
Joined: 26 Dec 2011
Posts: 114
Schools: HBS '18, IIMA
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

17 Jun 2015, 07:08
The sequence S is defined as follows for all n ≥ 1:

$$S_n=(-1)^n*\frac{1}{n(n+1)}$$

The sum of the first 10 terms of S is:

(A) Between –1 and –1/2
(B) Between –1/2 and 0
(C) Between 0 and 1/2
(D) Between 1/2 and 1
(E) Greater than 1

Solution -

S1 = -1/2
S2 = 1/6
S3 = -1/12
S4 = 1/20 .....

Neglecting the remaining terms as fractions does not vary the sum more than 1%. [We do not have time to calculate all in GMAT test]

S=S1+S2+S3+S4...S10 > -1/3, but S<-1/4 .

-1/3 <S< -1/4. So answer falls in B.

Thanks,

_________________

Thanks,

CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2788
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

17 Jun 2015, 07:18
balamoon wrote:
The sequence S is defined as follows for all n ≥ 1:

$$S_n=(-1)^n*\frac{1}{n(n+1)}$$

The sum of the first 10 terms of S is:

(A) Between –1 and –1/2
(B) Between –1/2 and 0
(C) Between 0 and 1/2
(D) Between 1/2 and 1
(E) Greater than 1

Solution -

S1 = -1/2
S2 = 1/6
S3 = -1/12
S4 = 1/20 .....

Neglecting the remaining terms as fractions does not vary the sum more than 1%. [We do not have time to calculate all in GMAT test]

S=S1+S2+S3+S4...S10 > -1/3, but S<-1/4 .

-1/3 <S< -1/4
. So answer falls in B.

Thanks,

I think the highlighted part in your above mentioned explanation is INCORRECT.

The actual Sum of the series is -0.3821789 which does NOT satisfy the range -1/3 <S< -1/4
_________________

Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION

Manager
Joined: 17 Mar 2014
Posts: 227
Location: India
Concentration: Operations, Strategy
GMAT 1: 670 Q48 V35
GPA: 3.19
WE: Information Technology (Computer Software)
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

17 Jun 2015, 10:45
1
lets check first 4 terms only ,
S1 = -1/2
S2 = 1/6
S3 = -1/12
S4 = 1/20

Add S1 and S2 , = -1/3
So sum of every term in series is negative so sum of entire first 10 term is negative ... this eliminates option C, D , E

Now S1+S2 = -1/3 = -0.333
after this terms are only going get smaller and smaller . The sum of next 8 terms can not shoot this value beyond -0.50 because then in that case sum (s3 +s4+ ..... +s10) should constiute more than 30% of (S1 to S10) , which looks imposible as values will get smaller and smaller only .

So total should lie between 0 and -1/2.
_________________

Press +1 Kudos if you find this Post helpful

Math Expert
Joined: 02 Sep 2009
Posts: 53020
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

22 Jun 2015, 05:56
Bunuel wrote:
The sequence S is defined as follows for all n ≥ 1:

$$S_n=(-1)^n*\frac{1}{n(n+1)}$$

The sum of the first 10 terms of S is:

(A) Between –1 and –1/2
(B) Between –1/2 and 0
(C) Between 0 and 1/2
(D) Between 1/2 and 1
(E) Greater than 1

MANHATTAN GMAT OFFICIAL SOLUTION:

We should compute the first few elements of Sn. Because we need to know the sum of the first 10 elements, we should also track the cumulative sum:

By now (if not before!) the pattern should be fairly obvious. The sum of the first n terms of Sn converges somewhere in the range between –0.3667 and –0.4. Only (B) exhibits a range in which the sum of this series could converge.

Attachment:

2015-06-22_1755.png [ 99.92 KiB | Viewed 1948 times ]

_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 9869
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the  [#permalink]

### Show Tags

06 Feb 2019, 10:07
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.
_________________
Re: The sequence S is defined as follows for all n ≥ 1: The sum of the   [#permalink] 06 Feb 2019, 10:07
Display posts from previous: Sort by