Find all School-related info fast with the new School-Specific MBA Forum

It is currently 17 Sep 2014, 13:41

Close

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
Your Progress

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Math : Sequences & Progressions

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
31 KUDOS received
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 76

Kudos [?]: 480 [31] , given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Math : Sequences & Progressions [#permalink] New post 28 Sep 2010, 15:32
31
This post received
KUDOS
13
This post was
BOOKMARKED
Sequences & Progressions
Image

This post is a part of [GMAT MATH BOOK]

created by: shrouded1


Definition

Sequence : It is an ordered list of objects. It can be finite or infinite. The elements may repeat themselves more than once in the sequence, and their ordering is important unlike a set

Arithmetic Progressions

Definition
It is a special type of sequence in which the difference between successive terms is constant.

General Term
a_n = a_{n-1} + d = a_1 + (n-1)d
a_i is the ith term
d is the common difference
a_1 is the first term

Defining Properties
Each of the following is necessary & sufficient for a sequence to be an AP :
  • a_i - a_{i-1} = Constant
  • If you pick any 3 consecutive terms, the middle one is the mean of the other two
  • For all i,j > k >= 1 : \frac{a_i - a_k}{i-k} = \frac{a_j-a_k}{j-k}

Summation
The sum of an infinite AP can never be finite except if a_1=0 & d=0
The general sum of a n term AP with common difference d is given by \frac{n}{2}(2a+(n-1)d)
The sum formula may be re-written as n * Avg(a_1,a_n) = \frac{n}{2} * (FirstTerm+LastTerm)

Examples
  1. All odd positive integers : {1,3,5,7,...} a_1=1, d=2
  2. All positive multiples of 23 : {23,46,69,92,...} a_1=23, d=23
  3. All negative reals with decimal part 0.1 : {-0.1,-1.1,-2.1,-3.1,...} a_1=-0.1, d=-1

Geometric Progressions

Definition
It is a special type of sequence in which the ratio of consequetive terms is constant

General Term
b_n = b_{n-1} * r = a_1 * r^{n-1}
b_i is the ith term
r is the common ratio
b_1 is the first term

Defining Properties
Each of the following is necessary & sufficient for a sequence to be an AP :
  • \frac{b_i}{b_{i-1}} = Constant
  • If you pick any 3 consecutive terms, the middle one is the geometric mean of the other two
  • For all i,j > k >= 1 : (\frac{b_i}{b_k})^{j-k} = (\frac{b_j}{b_k})^{i-k}

Summation
The sum of an infinite GP will be finite if absolute value of r < 1
The general sum of a n term GP with common ratio r is given by b_1*\frac{r^n - 1}{r-1}
If an infinite GP is summable (|r|<1) then the sum is \frac{b_1}{1-r}

Examples
  1. All positive powers of 2 : {1,2,4,8,...} b_1=1, r=2
  2. All negative powers of 4 : {1/4,1/16,1/64,1/256,...} b_1=1/4, r=1/4, sum=\frac{1/4}{(1-1/4)}=(1/3)

Harmonic Progressions

Definition
It is a special type of sequence in which if you take the inverse of every term, this new sequence forms an AP

Important Properties
Of any three consecutive terms of a HP, the middle one is always the harmonic mean of the other two, where the harmonic mean (HM) is defined as :
\frac{1}{2} * (\frac{1}{a} + \frac{1}{b}) = \frac{1}{HM(a,b)}
Or in other words :
HM(a,b) = \frac{2ab}{a+b}

APs, GPs, HPs : Linkage

Each progression provides us a definition of "mean" :

Arithmetic Mean : \frac{a+b}{2} OR \frac{a1+..+an}{n}
Geometric Mean : \sqrt{ab} OR (a1 *..* an)^{\frac{1}{n}}
Harmonic Mean : \frac{2ab}{a+b} OR \frac{n}{\frac{1}{a1}+..+\frac{1}{an}}

For all non-negative real numbers : AM >= GM >= HM

In particular for 2 numbers : AM * HM = GM * GM

Example :
Let a=50 and b=2,
then the AM = (50+2)*0.5 = 26 ;
the GM = sqrt(50*2) = 10 ;
the HM = (2*50*2)/(52) = 3.85
AM > GM > HM
AM*HM = 100 = GM^2

Misc Notes
A subsequence (any set of consequutive terms) of an AP is an AP

A subsequence (any set of consequutive terms) of a GP is a GP

A subsequence (any set of consequutive terms) of a HP is a HP

If given an AP, and I pick out a subsequence from that AP, consisting of the terms a_{i1},a_{i2},a_{i3},... such that i1,i2,i3 are in AP then the new subsequence will also be an AP

For Example : Consider the AP with a_1=1, d=2 {1,3,5,7,9,11,...}, so a_n=1+2*(n-1)=2n-1
Pick out the subsequence of terms a_5,a_{10},a_{15},...
New sequence is {9,19,29,...} which is an AP with a_1=9 and d=10

If given a GP, and I pick out a subsequence from that GP, consisting of the terms b_{i1},b_{i2},b_{i3},... such that i1,i2,i3 are in AP then the new subsequence will also be a GP


For Example : Consider the GP with b_1=1, r=2 {1,2,4,8,16,32,...}, so b_n=2^(n-1)
Pick out the subsequence of terms b_2,b_4,b_6,...
New sequence is {4,16,64,...} which is a GP with b_1=4 and r=4

The special sequence in which each term is the sum of previous two terms is known as the fibonacci sequence. It is neither an AP nor a GP. The first two terms are 1. {1,1,2,3,5,8,13,...}

In a finite AP, the mean of all the terms is equal to the mean of the middle two terms if n is even and the middle term if n is even. In either case this is also equal to the mean of the first and last terms

Some examples

Example 1
A coin is tossed repeatedly till the result is a tails, what is the probability that the total number of tosses is less than or equal to 5 ?

Solution
P(<=5 tosses) = P(1 toss)+...+P(5 tosses) = P(T)+P(HT)+P(HHT)+P(HHHT)+P(HHHHT)
We know that P(H)=P(T)=0.5
So Probability = 0.5 + 0.5^2 + ... + 0.5^5
This is just a finite GP, with first term = 0.5, n=5 and ratio = 0.5. Hence :
Probability = 0.5 * \frac{1-0.5^5}{1-0.5} = \frac{1}{2} * \frac{\frac{31}{32}}{\frac{1}{2}} = \frac{31}{32}

Example 2
In an arithmetic progression a1,a2,...,a22,a23, the common difference is non-zero, how many terms are greater than 24 ?
(1) a1 = 8
(2) a12 = 24

Solution
(1) a1=8, does not tell us anything about the common difference, so impossible to say how many terms are greater than 24
(2) a12=24, and we know common difference is non-zero. So either all the terms below a12 are greater than 24 and the terms above it less than 24 or the other way around. In either case, there are exactly 11 terms either side of a12. Sufficient
Answer is B

Example 3
For positive integers a,b (a<b) arrange in ascending order the quantities a, b, sqrt(ab), avg(a,b), 2ab/(a+b)

Solution
Using the inequality AM>=GM>=HM, the solution is :
a <= 2ab/(a+b) <= Sqrt(ab) <= Avg(a,b) <= b

Example 4
For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) *(1/2^k). If T is the sum of the first 10 terms in the sequence then T is

a)greater than 2
b)between 1 and 2
c)between 1/2 and 1
d)between 1/4 and 1/2
e)less than 1/4.

Solution
The sequence given has first term 1/2 and each subsequent term can be obtained by multiplying with -1/2. So it is a GP. We can use the GP summation formula
S=b\frac{1-r^n}{1-r}=\frac{1}{2} * \frac{1-(-1/2)^{10}}{1-(-1/2)} = \frac{1}{3} * \frac{1023}{1024}
1023/1024 is very close to 1, so this sum is very close to 1/3
Answer is d

Example 5
The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression?
A. 300
B. 120
C. 150
D. 170
E. 270

Solution
a_4+a_12=20
a_4=a_1+3d, a_12=a_1+11d
2a_1+14d=20
Now we need the sum of first 15 terms, which is given by :
\frac{15}{2} (2a_1 + (15-1)d) = \frac{15}{2} * (2a_1+14d) = 150
Answer is (c)

Additional Exercises


_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership


Last edited by Bunuel on 02 May 2014, 01:48, edited 9 times in total.
added some more points
Kaplan GMAT Prep Discount CodesKnewton GMAT Discount CodesManhattan GMAT Discount Codes
1 KUDOS received
CEO
CEO
User avatar
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2793
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 178

Kudos [?]: 948 [1] , given: 235

GMAT Tests User Reviews Badge
Re: Math : Sequences & Progressions [#permalink] New post 28 Sep 2010, 16:25
1
This post received
KUDOS
Great initiative.

Suggestions :

1. Include AM, GM , HM included between 2 numbers.
2. use a_{n} with math tag to get a_{n}
3. Include more examples.

This post,along with the algebra, is a good initiative to fill the important topics of Math Book that were not included earlier.
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned :)

Jo Bole So Nihaal , Sat Shri Akaal

:thanks Support GMAT Club by putting a GMAT Club badge on your blog/Facebook :thanks

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Gmat test review :
670-to-710-a-long-journey-without-destination-still-happy-141642.html

Intern
Intern
avatar
Joined: 26 Sep 2010
Posts: 16
Location: Singapore
Schools: Insead,NTU,NUS
WE 1: 6.5 years in Decision Sciences
Followers: 1

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

Re: Math : Sequences & Progressions [#permalink] New post 28 Sep 2010, 21:33
Thanks for putting it here!
2 KUDOS received
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 76

Kudos [?]: 480 [2] , given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: Math : Sequences & Progressions [#permalink] New post 29 Sep 2010, 09:27
2
This post received
KUDOS
gurpreetsingh wrote:
Great initiative.

Suggestions :

1. Include AM, GM , HM included between 2 numbers.
2. use a_{n} with math tag to get a_{n}
3. Include more examples.

This post,along with the algebra, is a good initiative to fill the important topics of Math Book that were not included earlier.


Check
Check
Check

Let me know what else ?
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Senior Manager
Senior Manager
User avatar
Status: Time to step up the tempo
Joined: 24 Jun 2010
Posts: 410
Location: Milky way
Schools: ISB, Tepper - CMU, Chicago Booth, LSB
Followers: 7

Kudos [?]: 117 [0], given: 50

Re: Math : Sequences & Progressions [#permalink] New post 29 Sep 2010, 18:30
Great initiative. +1 to you.
_________________

:good Support GMAT Club by putting a GMAT Club badge on your blog :thanks

Manager
Manager
avatar
Joined: 16 Aug 2009
Posts: 222
Followers: 3

Kudos [?]: 14 [0], given: 18

GMAT Tests User
Re: Math : Sequences & Progressions [#permalink] New post 30 Sep 2010, 04:25
Great one :)
Kudos to you !
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 76

Kudos [?]: 480 [0], given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: Math : Sequences & Progressions [#permalink] New post 07 Oct 2010, 00:33
added a couple more solved GMAT style questions to the end
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Senior Manager
Senior Manager
avatar
Joined: 31 Mar 2010
Posts: 415
Location: Europe
Followers: 2

Kudos [?]: 35 [0], given: 26

GMAT Tests User
Re: Math : Sequences & Progressions [#permalink] New post 07 Oct 2010, 00:56
Good to know. It can definitely same valuable time.
Manager
Manager
avatar
Joined: 19 Apr 2010
Posts: 216
Schools: ISB, HEC, Said
Followers: 4

Kudos [?]: 18 [0], given: 28

GMAT Tests User
Re: Math : Sequences & Progressions [#permalink] New post 17 Oct 2010, 23:10
This is great post but I was just wondering whether we need to know these concepts for GMAT
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 76

Kudos [?]: 480 [0], given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: Math : Sequences & Progressions [#permalink] New post 18 Oct 2010, 22:49
There are several questions in the OG that use these concepts. So I think its good to know all this. Plus if you search through the forums you'll find several Qs on sequences and progressions as well
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Senior Manager
Senior Manager
User avatar
Joined: 20 Jan 2010
Posts: 278
Schools: HBS, Stanford, Haas, Ross, Cornell, LBS, INSEAD, Oxford, IESE/IE
Followers: 14

Kudos [?]: 138 [0], given: 117

GMAT Tests User
Re: Math : Sequences & Progressions [#permalink] New post 29 Oct 2010, 20:19
Valuable resource.
Kudos +1 :)
_________________

"Don't be afraid of the space between your dreams and reality. If you can dream it, you can make it so."
Target=780
http://challengemba.blogspot.com
Kudos??

3 KUDOS received
Intern
Intern
avatar
Joined: 14 Oct 2010
Posts: 1
Followers: 0

Kudos [?]: 3 [3] , given: 1

Re: Math : Sequences & Progressions [#permalink] New post 02 Nov 2010, 10:53
3
This post received
KUDOS
I had a question. I am not sure if this works: "In case of n numbers : AM * HM = GM^n".

This seems to work for 2 numbers but for more than 2, it seems to break, please let me know if I am missing something.
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 76

Kudos [?]: 480 [0], given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: Math : Sequences & Progressions [#permalink] New post 02 Nov 2010, 11:28
nitantsharma wrote:
I had a question. I am not sure if this works: "In case of n numbers : AM * HM = GM^n".

This seems to work for 2 numbers but for more than 2, it seems to break, please let me know if I am missing something.


You are correct, this should only hold for special case n=2. Thanks for pointing out
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Senior Manager
Senior Manager
avatar
Joined: 08 Jun 2010
Posts: 454
Followers: 0

Kudos [?]: 28 [0], given: 39

GMAT Tests User
Re: Math : Sequences & Progressions [#permalink] New post 11 Nov 2010, 20:52
How can we get the formular for sumation of GEOMETRIC PROGRESSION. Please, prove, so that I do not have to remember the formular but to know the way to get the formular and so can solve the relative questions.
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 76

Kudos [?]: 480 [0], given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: Math : Sequences & Progressions [#permalink] New post 12 Nov 2010, 01:34

!
This proof is beyond the scope of the GMAT


The proof below is based on mathematical induction

To prove : The sum of an n term GP : b,br,br^2,...,br^{n-1} is b*\frac{r^n-1}{r-1}

P(1 term) : The sum of the GP {b} is b*\frac{r^1-1}{r-1}=b. Which is true trivially

P(n terms) : Let the sum of an n term GP : b,br,...,br^{n-1} be b*\frac{r^n-1}{r-1}

P(n+1 terms) : Consider the n+1 term GP : b,br,....,br^n
Sum of this GP = Sum of n term GP + br^n = b*\frac{r^n-1}{r-1} + br^n
Sum = \frac{b}{r-1} * (r^n - 1 + r^n(r-1))
=\frac{b}{r-1} *(r^{n+1}-1)

Hence P(1) is true
And if we assume P(n) true P(n+1) is true
By mathematical induction P(k) must be true for all k>=1
Hence, proved
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Director
Director
avatar
Joined: 23 Apr 2010
Posts: 583
Followers: 2

Kudos [?]: 26 [0], given: 7

Re: Math : Sequences & Progressions [#permalink] New post 17 Jan 2011, 02:09
Can someone please clarify:

I think the formula for calculating the sum of n consecutive numbers should be:

(lastterm - firstterm)*(lastterm - firstterm + 1)/2
Senior Manager
Senior Manager
User avatar
Affiliations: SPG
Joined: 15 Nov 2006
Posts: 326
Followers: 11

Kudos [?]: 266 [0], given: 20

GMAT Tests User
Re: Math : Sequences & Progressions [#permalink] New post 24 Jan 2011, 09:48
nonameee wrote:
Can someone please clarify:

I think the formula for calculating the sum of n consecutive numbers should be:

(lastterm - firstterm)*(lastterm - firstterm + 1)/2


1,2,3,4,5

let's apply your formula on the above series.

\frac{(5-1)*(5-1+1)}{2} = \frac{4*5}{2} = 10

this is not correct. let's try another formula.

\frac{n}{2}(firstterm + lastterm)

\frac{5}{2}(5+1) = 15


\frac{1}{2}(firstterm + lastterm) basically gives you the avg of the series. when you multiply the avg with number of terms (n), you get the sum.

HTH

_________________

press kudos, if you like the explanation, appreciate the effort or encourage people to respond.

Download the Ultimate SC Flashcards

Director
Director
avatar
Joined: 23 Apr 2010
Posts: 583
Followers: 2

Kudos [?]: 26 [0], given: 7

Re: Math : Sequences & Progressions [#permalink] New post 25 Jan 2011, 01:49
dimitri92, thanks. I must have made a computational error.
Senior Manager
Senior Manager
User avatar
Joined: 08 Nov 2010
Posts: 422
WE 1: Business Development
Followers: 7

Kudos [?]: 34 [0], given: 161

GMAT ToolKit User GMAT Tests User
Re: Math : Sequences & Progressions [#permalink] New post 17 Feb 2011, 03:32
i know i read it somewhere, but i cant find it now. What are the ways to get the full Math book?
_________________

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Intern
Intern
avatar
Joined: 29 May 2011
Posts: 5
Schools: HBS
Followers: 0

Kudos [?]: 6 [0], given: 0

Re: Math : Sequences & Progressions [#permalink] New post 30 May 2011, 09:25
The general sum of a n term GP with common ratio r is given by


For sum of a GP:
When r > 1, the denominator is (r-1)
When r < 1, the denominator is (1-r)
Re: Math : Sequences & Progressions   [#permalink] 30 May 2011, 09:25
    Similar topics Author Replies Last post
Similar
Topics:
Experts publish their posts in the topic Progress! transfer9858 2 22 Mar 2012, 13:52
Progression ISB2011 1 25 Apr 2010, 11:22
Progress? josh478 6 05 Nov 2006, 18:13
Progress OasisNYK 5 29 May 2006, 06:43
Sequences w07 7 16 Jul 2005, 06:45
Display posts from previous: Sort by

Math : Sequences & Progressions

  Question banks Downloads My Bookmarks Reviews Important topics  

Go to page    1   2   3    Next  [ 44 posts ] 



GMAT Club MBA Forum Home| About| Privacy Policy| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

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®.