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

 It is currently 16 Oct 2019, 00:22 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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  In the sequence a1,a2,a3...an, each term an is defined as an

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

Hide Tags

Intern  Joined: 25 Aug 2014
Posts: 6
In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

1
1
10 00:00

Difficulty:   55% (hard)

Question Stats: 64% (02:08) correct 36% (02:10) wrong based on 124 sessions

HideShow timer Statistics

In the sequence a1,a2,a3...an, each term an is defined as an=1/(−n)^n. The sum of the first ten terms of the sequence (a1,a2,a3...a10) must be __________.

A) Less than −1
B) Between −1 and −1/2
C) Between −1/2 and 0
D) Between 0 and 1/2
E) Greater than 1/2
Intern  Joined: 25 Aug 2014
Posts: 6
Re: In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

Can anyone explain better than the solution below?

As with most sequence problems, the biggest step to take is to jot out the first few terms in search of a pattern or some other insight about the way the sequence unfolds. Here, that gives you:

a1=1/(−1)^1=−1

a2=1/(−2)^2=1/4

a3=1/(−3)^3=−1/27

a4=1/(−4)^4=1/256
What you should see here is that the odd terms are all negative and the evens are positive, and that the terms get so small so quickly as to be inconsequential. After just adding the first few terms you should see that you have a little less than −3/4 and that the remaining terms are going to be too small to swing that value very much in either direction, so the answer must be B.
Intern  Joined: 27 Aug 2014
Posts: 27
GMAT Date: 09-27-2014
Re: In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

I think the approach is fine. This question is more about understanding that the increasing squares in the denominator make the terms relatively inconsequential to the previous term - with respect to the answer choices.
Intern  Joined: 27 Aug 2014
Posts: 5
Re: In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

ssagar wrote:
Can anyone explain better than the solution below?

As with most sequence problems, the biggest step to take is to jot out the first few terms in search of a pattern or some other insight about the way the sequence unfolds. Here, that gives you:

a1=1/(−1)^1=−1

a2=1/(−2)^2=1/4

a3=1/(−3)^3=−1/27

a4=1/(−4)^4=1/256
What you should see here is that the odd terms are all negative and the evens are positive, and that the terms get so small so quickly as to be inconsequential. After just adding the first few terms you should see that you have a little less than −3/4 and that the remaining terms are going to be too small to swing that value very much in either direction, so the answer must be B.

For those kinds of questions, I think the sum of the first 3 number will be sufficient to estimate the real sum.
The explanation for the method should be based on Taylor's theorem ( in my opinion)

I think for kinds of questions,
Intern  Joined: 07 Jan 2016
Posts: 21
Schools: AGSM '18
Re: In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

Somebody could give me the clear explanation?

I do not understand much about some ideas above.

Thank you.
Math Expert V
Joined: 02 Aug 2009
Posts: 7959
In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

oanhnguyen1116 wrote:
Somebody could give me the clear explanation?

I do not understand much about some ideas above.

Thank you.

Hi,
Since there has been some confusion till now on this Q, let me give you a method
one approach that can be useful is--

Quote:
In the sequence a1,a2,a3...an, each term an is defined as an=1/(−n)^n. The sum of the first ten terms of the sequence (a1,a2,a3...a10) must be __________.

A) Less than −1
B) Between −1 and −1/2
C) Between −1/2 and 0
D) Between 0 and 1/2
E) Greater than 1/2

Solution--

$$a_n=\frac{1}{{-n}^n}$$

so
1) $$a_1=\frac{1}{{-1}^1}$$
2) $$a_2=\frac{1}{{-2}^2}$$..
3) $$a_3=\frac{1}{{-3}^3}$$...
4) $$a_4=\frac{1}{{-4}^4}$$..

INFERENCE:-

1)Starting from -1, alternate terms are -ive and positive..
2) Numeric value keeps decreasing as we go up..

so two scenarios--

1) (first term + second term) + (third term+fourth term) +... (ninth+tenth)..
=> $$(-1+\frac{1}{4}) + (-\frac{1}{3^3} + \frac{1}{4^4}) ....+ (-\frac{1}{9^9} + \frac{1}{10^10})$$..
$$-\frac{3}{4} + (- ..) + (- ..)=- \frac{3}{4} - .. - ......$$
keeping the inference in mind, each pair will result in a negative qty
so the SUM will be lesser than $$-\frac{3}{4}$$..

2) lets take first term and then pair of 2nd-3rd, 4th-5th.. and so on
first term + (second term+third term)+(fourth term+fifth term)...(eighth+ninth)+tenth..
=>$$-1+ (\frac{1}{4}-\frac{1}{3^3} ) + ( \frac{1}{4^4}-..) .... (\frac{1}{8^8}-\frac{1}{9^9}) + \frac{1}{10^10}$$..
so leaving -1 and $$\frac{1}{10^10}$$, all pair will result in a positive value..
so$$-1+\frac{1}{10^10} +$$ some positive value + some positive value +...
so SUM has to be >-1..

Now we have two limits for the SUM..
$$SUM<-\frac{3}{4}$$and$$Sum>-1$$..
so it lies between -1 and -3/4..
choice B contains this RANGE..
so ans B. Between −1 and −1/2

_________________
Current Student D
Joined: 12 Aug 2015
Posts: 2569
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

chetan2u wrote:
oanhnguyen1116 wrote:
Somebody could give me the clear explanation?

I do not understand much about some ideas above.

Thank you.

Hi,
Since there has been some confusion till now on this Q, let me give you a method
one approach that can be useful is--

Quote:
In the sequence a1,a2,a3...an, each term an is defined as an=1/(−n)^n. The sum of the first ten terms of the sequence (a1,a2,a3...a10) must be __________.

A) Less than −1
B) Between −1 and −1/2
C) Between −1/2 and 0
D) Between 0 and 1/2
E) Greater than 1/2

Solution--

$$a_n=\frac{1}{{-n}^n}$$

so
1) $$a_1=\frac{1}{{-1}^1}$$
2) $$a_2=\frac{1}{{-2}^2}$$..
3) $$a_3=\frac{1}{{-3}^3}$$...
4) $$a_4=\frac{1}{{-4}^4}$$..

INFERENCE:-

1)Starting from -1, alternate terms are -ive and positive..
2) Numeric value keeps decreasing as we go up..

so two scenarios--

1) (first term + second term) + (third term+fourth term) +... (ninth+tenth)..
=> $$(-1+\frac{1}{4}) + (-\frac{1}{3^3} + \frac{1}{4^4}) ....+ (-\frac{1}{9^9} + \frac{1}{10^10})$$..
$$-\frac{3}{4} + (- ..) + (- ..)=- \frac{3}{4} - .. - ......$$
keeping the inference in mind, each pair will result in a negative qty
so the SUM will be lesser than $$-\frac{3}{4}$$..

2) lets take first term and then pair of 2nd-3rd, 4th-5th.. and so on
first term + (second term+third term)+(fourth term+fifth term)...(eighth+ninth)+tenth..
=>$$-1+ (\frac{1}{4}-\frac{1}{3^3} ) + ( \frac{1}{4^4}-..) .... (\frac{1}{8^8}-\frac{1}{9^9}) + \frac{1}{10^10}$$..
so leaving -1 and $$\frac{1}{10^10}$$, all pair will result in a positive value..
so$$-1+\frac{1}{10^10} +$$ some positive value + some positive value +...
so SUM has to be >-1..

Now we have two limits for the SUM..
$$SUM<-\frac{3}{4}$$and$$Sum>-1$$..
so it lies between -1 and -3/4..
choice B contains this RANGE..
so ans B. Between −1 and −1/2

Fab work...
I too did it the same way and came to the conclusion that the sum must be between -1 and -3/4 . Though i must confess the options bamboozled me a bit...
_________________
Intern  Joined: 07 Jan 2016
Posts: 21
Schools: AGSM '18
Re: In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

Thank you chetan2u so much! It's clear for me now <3
Manager  B
Joined: 03 Sep 2018
Posts: 174
Re: In the sequence a1,a2,a3...an, each term an is defined as an  [#permalink]

Show Tags

why can we not do 10(a_1+a_n)/2 ? Re: In the sequence a1,a2,a3...an, each term an is defined as an   [#permalink] 04 Oct 2019, 08:02
Display posts from previous: Sort by

In the sequence a1,a2,a3...an, each term an is defined as an

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

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