Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 24 May 2017, 16:35

### 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

# In a certain sequence of numbers, a1, a2, a3, ..., an, the

Author Message
TAGS:

### Hide Tags

Manager
Joined: 08 Dec 2012
Posts: 66
Location: United Kingdom
WE: Engineering (Consulting)
Followers: 2

Kudos [?]: 213 [2] , given: 31

In a certain sequence of numbers, a1, a2, a3, ..., an, the [#permalink]

### Show Tags

06 Oct 2013, 10:30
2
KUDOS
9
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

59% (02:45) correct 41% (01:39) wrong based on 203 sessions

### HideShow timer Statistics

In a certain sequence of numbers, a1, a2, a3, ..., an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1
[Reveal] Spoiler: OA
Intern
Joined: 06 Sep 2013
Posts: 16
Followers: 0

Kudos [?]: 13 [5] , given: 3

Re: In a certain sequence of numbers, a1,a2,a3,...,an, the avera [#permalink]

### Show Tags

06 Oct 2013, 10:45
5
KUDOS
nave81 wrote:
In a certain sequence of numbers, a1,a2,a3,...,an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

I got D: 19.

As stated above, (a1+a2+a3......+a10)/10=10
therefore a1+a2+a3.......a10=100 (1)
using the same logic, we got a1+a2+a3..........+a9=81 (2)
(2)-(1) we got a10=19
Math Expert
Joined: 02 Sep 2009
Posts: 38858
Followers: 7728

Kudos [?]: 106046 [9] , given: 11607

Re: In a certain sequence of numbers, a1,a2,a3,...,an, the avera [#permalink]

### Show Tags

06 Oct 2013, 10:49
9
KUDOS
Expert's post
1
This post was
BOOKMARKED
nave81 wrote:
In a certain sequence of numbers, a1,a2,a3,...,an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

The average of the first 9 consecutive terms starting with a1 is 9 --> $$\frac{a_1+a_2+a_3+...+a_{9}}{9}=9$$ --> $$a_1+a_2+a_3+...+a_{9}=81$$.

The average of the first 10 consecutive terms starting with a1 is 10 --> $$\frac{a_1+a_2+a_3+...+a_{10}}{10}=10$$ --> $$a_1+a_2+a_3+...+a_{10}=100$$.

Subtract the first equation from the second: $$a_{10}=100-81=19$$.

_________________
Manager
Joined: 29 Aug 2013
Posts: 77
Location: United States
GMAT 1: 590 Q41 V29
GMAT 2: 540 Q44 V20
GPA: 3.5
WE: Programming (Computer Software)
Followers: 0

Kudos [?]: 64 [1] , given: 24

Re: In a certain sequence of numbers, a1,a2,a3,...,an, the avera [#permalink]

### Show Tags

06 Oct 2013, 22:42
1
KUDOS
nave81 wrote:
In a certain sequence of numbers, a1,a2,a3,...,an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

The average of 1st 10 numbers is 10

Hence,
The average of 5th and 6th term is 10 and therefore 5th term is 9 and 6th term is 11
Extrapolating this trend we get 10th term = 19 i.e. 9,11,13,15,17,19
VP
Status: Been a long time guys...
Joined: 03 Feb 2011
Posts: 1381
Location: United States (NY)
Concentration: Finance, Marketing
GPA: 3.75
Followers: 178

Kudos [?]: 1462 [4] , given: 62

Re: In a certain sequence of numbers, a1,a2,a3,...,an, the avera [#permalink]

### Show Tags

06 Oct 2013, 23:17
4
KUDOS
nave81 wrote:
In a certain sequence of numbers, a1,a2,a3,...,an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

A slight different approach.
Sum of first m numbers is: $$[(First term$$+$$Last term)$$$$/$$$$2]$$*$$m$$.
Here it will be $$(a1+a10)/2$$*$$m$$.
Also, since the average is given as m, therefore the sum would be $$m^2$$.

On solving, $$a1+a10=2m$$
$$a1+a10=20$$ or $$a10=20-1=19$$
_________________
Intern
Joined: 21 Sep 2013
Posts: 30
Location: United States
Concentration: Finance, General Management
GMAT Date: 10-25-2013
GPA: 3
WE: Operations (Mutual Funds and Brokerage)
Followers: 0

Kudos [?]: 25 [0], given: 82

Re: In a certain sequence of numbers, a1,a2,a3,...,an, the avera [#permalink]

### Show Tags

08 Oct 2013, 00:33
Is it appropriate to do some good guess work out by looking at the options?
For example if a1=1, and if a2, a3 ... a10 are in sequence then there are 3 possiblities for the answer of a10.
1st is that a1.. a10 are the first consecutive 10 nos. in that case the ans should be 10 which is not one of the options.
2nd is that a1... a10 could even consecutive nos. in that case nth term equals to 2n, so in our case 10 th term should be 2*10= 20. not one of the options.
3rd is that a1...a10 could be odd consecutive nos. in that case nth term equals to 2n-1, so in our case 10th term should be 2*10-1=19. Option D our ans.

Please correct me if my thought process is wrong.

Bunuel wrote:
nave81 wrote:
In a certain sequence of numbers, a1,a2,a3,...,an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

The average of the first 9 consecutive terms starting with a1 is 9 --> $$\frac{a_1+a_2+a_3+...+a_{9}}{9}=9$$ --> $$a_1+a_2+a_3+...+a_{9}=81$$.

The average of the first 10 consecutive terms starting with a1 is 10 --> $$\frac{a_1+a_2+a_3+...+a_{10}}{10}=10$$ --> $$a_1+a_2+a_3+...+a_{10}=100$$.

Subtract the first equation from the second: $$a_{10}=100-81=19$$.

Manager
Status: Do till 740 :)
Joined: 13 Jun 2011
Posts: 110
Concentration: Strategy, General Management
GMAT 1: 460 Q35 V20
GPA: 3.6
WE: Consulting (Computer Software)
Followers: 1

Kudos [?]: 13 [0], given: 19

Re: In a certain sequence of numbers, a1,a2,a3,...,an, the avera [#permalink]

### Show Tags

24 Oct 2013, 17:33
or a simpler way is this:
Since the mean of m consectuive terms is m
(a1+a10)/2=10

Solve for a10 as we know a1

1+a10=20
a10=19

Hope this helps.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7372
Location: Pune, India
Followers: 2287

Kudos [?]: 15099 [1] , given: 224

Re: In a certain sequence of numbers, a1,a2,a3,...,an, the avera [#permalink]

### Show Tags

24 Oct 2013, 22:00
1
KUDOS
Expert's post
nave81 wrote:
In a certain sequence of numbers, a1,a2,a3,...,an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

Or use pattern recognition to solve it.
a1 = 1
a2 = ? the mean of a1 and a2 must be 2. Since a1 is 1, a2 must be 3
a2 = 3
a3 = ? the mean of a1, a2, a3 = 3. Since a1 and a2 are 1 and 3, a3 must be 5 to give a mean of 3
a3 = 5
a4 = ? the mean of a1, a2, a3 and a4 is 4. Since we have 1, 3, 5 and we need a mean of 4, a4 must be 7 (use deviation from mean to figure out each of these in a couple of secs)
a4 = 7

We see the pattern: 1, 3, 5, 7 ...
nth term is given by 2n - 1.
a10 is 2*10 - 1 = 19
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Current Student
Joined: 06 Sep 2013
Posts: 2005
Concentration: Finance
Followers: 68

Kudos [?]: 643 [0], given: 355

Re: In a certain sequence of numbers, a1, a2, a3, ..., an, the [#permalink]

### Show Tags

18 Dec 2013, 17:42
nave81 wrote:
In a certain sequence of numbers, a1, a2, a3, ..., an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

Sum of 'm' terms / m = Average of m

Now, average of m is also = (L + 1)/2 * (m)
Where, L is the last number of the sequence and 'm' is the number of terms
So then we have that L = 2m-1

For a10 then, we would have 2(10)-1 = 19

Hope it helps
Cheers!
J
Current Student
Joined: 06 Sep 2013
Posts: 2005
Concentration: Finance
Followers: 68

Kudos [?]: 643 [0], given: 355

Re: In a certain sequence of numbers, a1, a2, a3, ..., an, the [#permalink]

### Show Tags

20 Feb 2014, 14:29
Instead of consecutive terms I would say evenly spaced set in the question. Anyway, from the example we know that they are talking about consecutive odd numbers (n^2 is the sum of the first n positive odd numbers) therefore, the largest number of the first 10 odd integers is 19.

Hope it helps
Cheers
J
GMAT Tutor
Joined: 20 Jul 2012
Posts: 25
GMAT 1: 780 Q50 V50
Followers: 2

Kudos [?]: 2 [0], given: 4

Re: In a certain sequence of numbers, a1, a2, a3, ..., an, the [#permalink]

### Show Tags

20 Feb 2014, 15:57
2 ways algebra focused and 1 way emphasizing pattern recognition:

1) by given formula:the mean of a1 is 1 the mean of a1+ a2 is 2 the mean of a1+a2+ a3 is 3 and so on. likewise for mean of numbers up toaX it is X.

if mean of numbers up to a10 is 10 then sum of numbers for a10 is 10*10. likewise, the sum of numbers for a9 is 9*9.

100 -81 is the final term: a10 =19

Formula for sum of number (( first + last)/2 )*m number of terms.

2) more elegant way:

we already know that a1 = 1 and formula for sum of numbers. ((a1 +a10)/2)*m

since average of numbers = m then sum of number = m*m

so we have an algebraic equivalency. we know a1= 1 , m= 10 thus only 1 variable to determine is a10.

3) pattern recognition

a1 = 1
a2 = ? since the mean of a1 and a2 is 2. Since a1 is 1, a2 must be 3
a2 = 3
a3 = ? since the mean of a1, a2, a3 = 3. Since a1 and a2 are 1 and 3, a3 must be 5 to give a mean of 3
a3 = 5

the pattern: 1, 3, 5,

nth term is given by 2n - 1.

a10 is 2*10 - 1 = 19

hope this helps. Thanks!
_________________

WWW.CLEARMOUNTAINPREP.COM

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15431
Followers: 649

Kudos [?]: 207 [0], given: 0

Re: In a certain sequence of numbers, a1, a2, a3, ..., an, the [#permalink]

### Show Tags

17 Sep 2015, 00:49
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.
_________________
Manager
Joined: 03 Apr 2013
Posts: 149
Followers: 4

Kudos [?]: 22 [0], given: 720

In a certain sequence of numbers, a1, a2, a3, ..., an, the [#permalink]

### Show Tags

02 Jul 2016, 02:11
nave wrote:
In a certain sequence of numbers, a1, a2, a3, ..., an, the average (arithmetic mean) of the first m consecutive terms starting with a1 is m, for any positive integer m. If a1=1, what is a10?

A. 100
B. 55
C. 21
D. 19
E. 1

Another way..
$$a_1 = 1 (1^2 - 0^2)$$
$$a_2 = 3 (2^2 - 1^2)$$
$$a_3 = 5 (3^2 - 2^2)$$

Thus..

$$a_{10} = 19 (10^2 - 9^2)$$

_________________

Spread some love..Like = +1 Kudos

In a certain sequence of numbers, a1, a2, a3, ..., an, the   [#permalink] 02 Jul 2016, 02:11
Similar topics Replies Last post
Similar
Topics:
3 The sequence a1, a2, a3,…, an is defined by an = an – 2 + an – 1 for a 3 28 Apr 2017, 15:17
5 For the infinite sequence of numbers a1, a2, a3, ..., an, ..., for all 6 16 Oct 2016, 13:07
38 A sequence of numbers a1, a2, a3,…. is defined as follows: a1 = 3, a2 5 06 Aug 2016, 06:03
15 The sequence of numbers a1, a2, a3, ..., an is defined by an 6 01 Mar 2017, 03:32
21 The sequence a1, a2, a3, ... ,an, ... is such that an = an-1 16 18 Apr 2017, 22:27
Display posts from previous: Sort by