It is currently 21 Nov 2017, 11:02

### 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 June
Open Detailed Calendar

# For all positive integers n, the nth term in sequence S_n is defined

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 42283

Kudos [?]: 132935 [0], given: 12391

For all positive integers n, the nth term in sequence S_n is defined [#permalink]

### Show Tags

10 Sep 2015, 00:30
00:00

Difficulty:

35% (medium)

Question Stats:

63% (01:15) correct 37% (01:13) wrong based on 137 sessions

### HideShow timer Statistics

For all positive integers n, the nth term in sequence $$S_n$$ is defined as follows:

$$S_n = (n!)^{-1}$$

The sum of the first six terms of $$S_n$$ is

(A) between 0 and ½
(B) between ½ and 1
(C) between 1 and 1½
(D) between 1½ and 2
(E) greater than 2

Kudos for a correct solution.
[Reveal] Spoiler: OA

_________________

Kudos [?]: 132935 [0], given: 12391

Manager
Joined: 07 Apr 2015
Posts: 180

Kudos [?]: 70 [1], given: 185

For all positive integers n, the nth term in sequence S_n is defined [#permalink]

### Show Tags

10 Sep 2015, 01:30
1
KUDOS
The sequence for the first six terms is: 1, 1/2, 1/6, 1/24, 1/120, 1/720

You can immediately rule out answer A, B and C as 1 + 1/2 alone will yield a result greater than 1,5.

Now the proceeding terms will get smaller and smaller and will not be 0,5 if summed, therefore the total sum has to be somewhere between 1,5 and 2. D

Kudos [?]: 70 [1], given: 185

Manager
Joined: 10 Aug 2015
Posts: 98

Kudos [?]: 84 [0], given: 20

Re: For all positive integers n, the nth term in sequence S_n is defined [#permalink]

### Show Tags

10 Sep 2015, 04:58
Solution : Multiply and divide the sum by 6!
1/720(720+360+120+30+6+1)==> clearly b/n 1.5-2

Option D

Kudos [?]: 84 [0], given: 20

Manager
Joined: 29 Jul 2015
Posts: 159

Kudos [?]: 191 [0], given: 59

Re: For all positive integers n, the nth term in sequence S_n is defined [#permalink]

### Show Tags

10 Sep 2015, 06:59
Bunuel wrote:
For all positive integers n, the nth term in sequence $$S_n$$ is defined as follows:

$$S_n = (n!)^{-1}$$

The sum of the first six terms of $$S_n$$ is

(A) between 0 and ½
(B) between ½ and 1
(C) between 1 and 1½
(D) between 1½ and 2
(E) greater than 2

Kudos for a correct solution.

The sum of sequencee of first 6 terms of $$S_n$$ will be

1+ $$\frac{1}{2!}$$ + $$\frac{1}{3!}$$ + $$\frac{1}{4!}$$ + $$\frac{1}{5!}$$ + $$\frac{1}{6!}$$

We do not actually need to solve for the sum.

The first two terms are 1 and 1/2. So the sum should be definitely greater than 1½ .
For the sum to be greater than or equal to 2, the sum of next fractions should be at least 1/2.
Now the succeeding fractions are less than 1/2, hence irrespective of the fraction, the sum will be less than 1/2.
So, sum of the sequence will be greater than 1½ but not more than 2.

Kudos [?]: 191 [0], given: 59

Manager
Joined: 03 May 2014
Posts: 70

Kudos [?]: 57 [1], given: 43

Concentration: Operations, Marketing
Schools: ISB '17 (D)
GMAT 1: 680 Q48 V34
GMAT 2: 700 Q49 V35
WE: Engineering (Energy and Utilities)
For all positive integers n, the nth term in sequence S_n is defined [#permalink]

### Show Tags

10 Sep 2015, 10:31
1
KUDOS
Bunuel wrote:
For all positive integers n, the nth term in sequence $$S_n$$ is defined as follows:

$$S_n = (n!)^{-1}$$

The sum of the first six terms of $$S_n$$ is

(A) between 0 and ½
(B) between ½ and 1
(C) between 1 and 1½
(D) between 1½ and 2
(E) greater than 2

Kudos for a correct solution.

from the sequence equation we can see first 6 terms to be $$1+ \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} +\frac{1}{720}$$
So from 1st two numbers we have sum 1.5 and adding other four will be give more than 1.5 so A, B , C gone.
From D and E we can easily select D as from last four terms we have $$\frac{1}{6} (1+ \frac{1}{4} + \frac{1}{20} +\frac{1}{120})$$ which is less than .5. Hence D
_________________

Beat verbal, Beat GMAT...
Trying Hard to do that.....

Kudos if my post helped you

Kudos [?]: 57 [1], given: 43

Math Expert
Joined: 02 Sep 2009
Posts: 42283

Kudos [?]: 132935 [0], given: 12391

Re: For all positive integers n, the nth term in sequence S_n is defined [#permalink]

### Show Tags

13 Sep 2015, 09:36
Bunuel wrote:
For all positive integers n, the nth term in sequence $$S_n$$ is defined as follows:

$$S_n = (n!)^{-1}$$

The sum of the first six terms of $$S_n$$ is

(A) between 0 and ½
(B) between ½ and 1
(C) between 1 and 1½
(D) between 1½ and 2
(E) greater than 2

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

First, list out the first few terms of $$S_n$$ to get a feeling for the sequence.

$$S_1 = (1!)^{-1} = \frac{1}{1!} = \frac{1}{1} = 1$$
$$S_2 = (2!)^{-1} = \frac{1}{2!} = \frac{1}{2}$$

The sum of just the first two terms, by the way, is 1½. Glancing ahead at the answers, we should already be able to rule out A, B, and C at this point. The terms never go negative, so the only question is whether the sum of the first six terms takes us above 2 or not.

The sum of the first six terms can be written this way:

$$Sum = 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6!$$
$$= 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720$$

Given how quickly the fractions shrink, we might guess at this point that the sum does not go above 2. However, we can make sure of this guess. Leaving out the first two terms (the 1 and the ½), we can just figure out the sum of the last four terms:

$$1/6 + 1/24 + 1/120 + 1/720$$
$$= 1/6(1 + 1/4 + 1/20 + 1/120)$$
$$= 1/6(120/120 + 30/120 + 6/120 + 1/120)$$
$$= 1/6(157/120)$$
$$= 157/720$$,

which is definitely less than 1/2. (If it were above 1/2, then the sum of the first six terms would be greater than 2.)

Thus, the sum of the first six terms is between 1½ and 2.

_________________

Kudos [?]: 132935 [0], given: 12391

Non-Human User
Joined: 09 Sep 2013
Posts: 15576

Kudos [?]: 283 [0], given: 0

Re: For all positive integers n, the nth term in sequence S_n is defined [#permalink]

### Show Tags

21 Aug 2017, 12:01
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.
_________________

Kudos [?]: 283 [0], given: 0

Re: For all positive integers n, the nth term in sequence S_n is defined   [#permalink] 21 Aug 2017, 12:01
Display posts from previous: Sort by

# For all positive integers n, the nth term in sequence S_n is defined

 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®.