GMAT Changed on April 16th - Read about the latest changes here

 It is currently 20 May 2018, 20:28

### 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

# M10-36

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 45213

### Show Tags

16 Sep 2014, 00:43
2
KUDOS
Expert's post
5
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

55% (01:06) correct 45% (01:17) wrong based on 107 sessions

### HideShow timer Statistics

For a certain sequence $$a_1=9^1$$ and $$a_n=(10^n - 1)^n$$, where $$n$$ is a positive integer. If S is the sum of $$n$$ terms of the sequence, what is the units digit of S?

(1) $$n$$ is even

(2) $$n$$ is prime
[Reveal] Spoiler: OA

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 45213

### Show Tags

16 Sep 2014, 00:43
1
KUDOS
Expert's post
Official Solution:

We have the sum of $$n$$ integers:

(..9) + (..1) + (..9) + (..1) + ...

Statement (1) by itself is sufficient. If $$n$$ is even then this sum ends with 0. If it's odd, then this sum ends with 9.

Statement (2) by itself is insufficient. (2 is prime and even, 3 is prime and odd).

_________________
Manager
Joined: 01 Jul 2009
Posts: 204

### Show Tags

27 Mar 2015, 09:06
2
KUDOS
Bunuel wrote:
Official Solution:

We have the sum of $$n$$ integers:

(..9) + (..1) + (..9) + (..1) + ...

Statement (1) by itself is sufficient. If $$n$$ is even then this sum ends with 0. If it's odd, then this sum ends with 9.

Statement (2) by itself is insufficient. (2 is prime and even, 3 is prime and odd).

I completely agree with the solution. However, the way this sum is depicted one can assume that n is more than 3, since the powers of 2 and 3 are clearly depicted. So if we assume that n is more than 3 the answer will change.

Maybe the sum should have been depicted as

9^1 + ... + (10^n - 1)^n

in this case one cannot assume that n is more than 3?

Cuz during the test I assumed that N is more than 3 and so I got an answer different from the official one.
_________________

Consider giving Kudos if you like the post.

Intern
Joined: 11 Nov 2014
Posts: 39
Concentration: Technology, Strategy
GMAT 1: 660 Q48 V31
GMAT 2: 720 Q50 V37
GPA: 3.6
WE: Consulting (Consulting)

### Show Tags

27 Jun 2015, 15:14
GMAT888

I´m with you, I got the same understanding from the way the question is written
Current Student
Joined: 29 Mar 2015
Posts: 44
Location: United States

### Show Tags

18 Jul 2015, 00:43
I guess we should assume that n cannot be 0 either?
Math Expert
Joined: 02 Aug 2009
Posts: 5779

### Show Tags

18 Jul 2015, 01:57
bluesquare wrote:
I guess we should assume that n cannot be 0 either?

Hi,
why can't n be 0?
n is even as per statement 1 and if it is to power 0,which is even, answer will still be 1(anything to pwer of 0 is 1)... suff
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

GMAT online Tutor

Math Expert
Joined: 02 Aug 2009
Posts: 5779

### Show Tags

18 Jul 2015, 02:02
1
KUDOS
Expert's post
michaelyb wrote:
GMAT888

I´m with you, I got the same understanding from the way the question is written

Hi,
I think the Q is correct as it is written if you take n to be a positive integer > or <3 will still give you the same answer...
Bunuel , only thing that could have been mentioned is n is a non negative integer... because nonnegative would change the answer
or we take this as a series and therefore n as positive integer but it is not mentioned that it is a sum of a series
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

GMAT online Tutor

Current Student
Joined: 29 Mar 2015
Posts: 44
Location: United States

### Show Tags

18 Jul 2015, 17:25
chetan2u wrote:
bluesquare wrote:
I guess we should assume that n cannot be 0 either?

Hi,
why can't n be 0?
n is even as per statement 1 and if it is to power 0,which is even, answer will still be 1(anything to pwer of 0 is 1)... suff

If n can be 0, then in statement 1, n is even means that the term can end in either a 1 or a 9, so statement 1 should be insufficient, no?
Math Expert
Joined: 02 Sep 2009
Posts: 45213

### Show Tags

20 Jul 2015, 05:00
Bunuel wrote:
For a certain sequence $$a_1=9^1$$ and $$a_n=(10^n - 1)^n$$, where $$n$$ is a positive integer. If S is the sum of $$n$$ terms of the sequence, what is the units digit of S?

(1) $$n$$ is even

(2) $$n$$ is prime

Edited the question. Is it better?
_________________
Manager
Joined: 10 Feb 2017
Posts: 51
Location: India
GMAT 1: 680 Q50 V30
GPA: 3.9

### Show Tags

24 Mar 2017, 01:38
please can someone elaborate on the answer and explanation i cannot understand why cannot n be greater than 3.

i tried forming series first as
9,99^2, 999^3, 9999^4...... (10^n - 1)^n

how come series is in form ..9,..1,..9,..1 ??? what is this clearly ??

Math Expert
Joined: 02 Sep 2009
Posts: 45213

### Show Tags

24 Mar 2017, 04:40
3
KUDOS
Expert's post
smanujahrc wrote:
For a certain sequence $$a_1=9^1$$ and $$a_n=(10^n - 1)^n$$, where $$n$$ is a positive integer. If S is the sum of $$n$$ terms of the sequence, what is the units digit of S?

please can someone elaborate on the answer and explanation i cannot understand why cannot n be greater than 3.

i tried forming series first as
9,99^2, 999^3, 9999^4...... (10^n - 1)^n

how come series is in form ..9,..1,..9,..1 ??? what is this clearly ??

That the terms of the sequence are given by the formula $$a_n=(10^n - 1)^n$$.

$$a_1=9^1=9$$
$$a_2=99^2=...1$$
$$a_3=999^3=...9$$
$$a_4=9999^4=...1$$
...

As you can see odd terms have the units digit of 9 and the even terms have the units digit of 1.

The question asks about the units digit of the sum of $$n$$ terms of the sequence. Now, if n is even then this sum ends with 0. If it's odd, then this sum ends with 9.

For example:
If n = 2, then the units digit of the sum of 2 terms is 9^1 + 99^2 = ...0.
If n = 3, then the units digit of the sum of 3 terms is 9^1 + 99^2 + 999^3 = ...9.

Statement (1) by itself is sufficient. n = 2 --> the sum ends with 0.

Statement (2) by itself is insufficient. (2 is prime and even, 3 is prime and odd).

Hope it's clear.
_________________
Intern
Joined: 05 Sep 2016
Posts: 22

### Show Tags

07 Feb 2018, 10:57
Bunuel wrote:
smanujahrc wrote:
For a certain sequence $$a_1=9^1$$ and $$a_n=(10^n - 1)^n$$, where $$n$$ is a positive integer. If S is the sum of $$n$$ terms of the sequence, what is the units digit of S?

please can someone elaborate on the answer and explanation i cannot understand why cannot n be greater than 3.

i tried forming series first as
9,99^2, 999^3, 9999^4...... (10^n - 1)^n

how come series is in form ..9,..1,..9,..1 ??? what is this clearly ??

That the terms of the sequence are given by the formula $$a_n=(10^n - 1)^n$$.

$$a_1=9^1=9$$
$$a_2=99^2=...1$$
$$a_3=999^3=...9$$
$$a_4=9999^4=...1$$
...

As you can see odd terms have the units digit of 9 and the even terms have the units digit of 1.

The question asks about the units digit of the sum of $$n$$ terms of the sequence. Now, if n is even then this sum ends with 0. If it's odd, then this sum ends with 9.

For example:
If n = 2, then the units digit of the sum of 2 terms is 9^1 + 99^2 = ...0.
If n = 3, then the units digit of the sum of 3 terms is 9^1 + 99^2 + 999^3 = ...9.

Statement (1) by itself is sufficient. n = 2 --> the sum ends with 0.

Statement (2) by itself is insufficient. (2 is prime and even, 3 is prime and odd).

Hope it's clear.

Brilliant explanation. Thanx a lot. Its is superb.
Re: M10-36   [#permalink] 07 Feb 2018, 10:57
Display posts from previous: Sort by

# M10-36

Moderators: chetan2u, Bunuel

 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®.