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

It is currently 20 Feb 2019, 09:46

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.

Close

Request Expert Reply

Confirm Cancel
Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
  • Free GMAT Prep Hour

     February 20, 2019

     February 20, 2019

     08:00 PM EST

     09:00 PM EST

    Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST
  • Online GMAT boot camp for FREE

     February 21, 2019

     February 21, 2019

     10:00 PM PST

     11:00 PM PST

    Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

Formulas for Consecutive, Even, Odd Integers

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

Hide Tags

 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 53020
Formulas for Consecutive, Even, Odd Integers  [#permalink]

Show Tags

New post 03 Feb 2019, 14:46
1
4
Formulas for Consecutive Integers, Even integers, Odd Integers, etc




Consecutive Integers

DEFINITION:
Consecutive integers are integers that follow one another, without skipping any integers. So, "consecutive integers" ALWAYS mean integers that follow each other in order with common difference of 1: ... x-3, x-2, x-1, x, x+1, x+2, ....
Examples:
7, 8, 9, and -2, -1, 0, 1, are consecutive integers.
2, 4, 6 ARE NOT consecutive integers, they are consecutive even integers.
3, 5, 7 ARE NOT consecutive integers, they are consecutive odd integers.

FORMULAS:
• Sum of \(n\) consecutive integers equals the mean multiplied by the number of terms, \(n\): \(Sum=mean*number \ of \ terms\).
Example: given consecutive integers \(\{-3, -2, -1, 0, 1,2\}\), \(mean=\frac{-3+2}{2}=-\frac{1}{2}\), (mean equals to the average of the first and last terms), so the sum equals to \(-\frac{1}{2}*6=-3\).

PROPERTIES:
• If n is odd, the sum of consecutive integers is always divisible by n.
Example:: given \(\{9,10,11\}\), we have \(n=3\) consecutive integers. The sum of 9+10+11=30, therefore, is divisible by 3.

• If n is even, the sum of consecutive integers is never divisible by n.
Example: given \(\{9,10,11,12\}\), we have \(n=4\) consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of \(n\) consecutive integers is always divisible by \(n!\).
Example: given \(n=4\) consecutive integers: \(\{3,4,5,6\}\). The product of 3*4*5*6 is 360, which is divisible by 4!=24.



Evenly Spaced Set

DEFINITION:
Evenly spaced set or an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
Examples: the set of integers \(\{9,13,17,21\}\) is an example of evenly spaced set with common difference of 4. Set of consecutive integers, {-4, -2, 0, 2, 4, 6, ...} is also an example of evenly spaced set with common difference of 2.

FORMULAS:
• If the first term is \(a_1\) and the common difference of successive members is \(d\), then the \(n_{th}\) term of the sequence is given by:
\(a_ n=a_1+d(n-1)\)

• In any evenly spaced set the arithmetic mean (average) is equal to the median and can be calculated by the formula \(mean=median=\frac{a_1+a_n}{2}\), where \(a_1\) is the first term and \(a_n\) is the last term.
Examples: given the set \(\{7,11,15,19\}\), \(mean=median=\frac{7+19}{2}=13\).

• The sum of the elements in any evenly spaced set is given by: \(Sum=\frac{a_1+a_n}{2}*n\), the mean multiplied by the number of terms. OR, \(Sum=\frac{2a_1+d(n-1)}{2}*n\)

• If the evenly spaced set contains odd number of elements, the mean is the middle term, so the sum is middle term multiplied by number of terms.
Example: There are five terms in the set {1, 7, 13, 19, 25}, middle term is 13, so the sum is 13*5 =65.


Formulas For Special Cases:

Sum of n first positive integers: \(1+2+...+n=\frac{1+n}{2}*n\)
Example: given \(n=4\) the sum of four first positive integers \(1+2+3+4=\frac{1+4}{2}*4=10\).

Sum of n first positive ODD numbers: \(a_1+a_2+...+a_n=1+3+...+a_n=n^2\), where \(a_n\) is the last, \(n_{th}\) term and given by: \(a_n=2n-1\).
Example: given \(n=5\) first odd positive integers, then their sum equals to \(1+3+5+7+9=5^2=25\).

Sum of n first positive EVEN numbers: \(a_1+a_2+...+a_n=2+4+...+a_n\)\(=n(n+1)\), where \(a_n\) is the last, \(n_{th}\) term and given by: \(a_n=2n\).
Example: given \(n=4\) first positive even integers, then their sum equals to \(2+4+6+8=4(4+1)=20\).



MORE THEORY AND QUESTIONS:



_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Director
Director
User avatar
G
Joined: 23 Feb 2015
Posts: 695
GMAT 1: 720 Q49 V40
GMAT ToolKit User Premium Member
Re: Formulas for Consecutive, Even, Odd Integers  [#permalink]

Show Tags

New post 03 Feb 2019, 14:59
Good initiative. Thanks bb.

Posted from my mobile device
_________________

“The heights by great men reached and kept were not attained in sudden flight but, they while their companions slept, they were toiling upwards in the night.”
― Henry Wadsworth Longfellow

Intern
Intern
avatar
B
Joined: 08 Feb 2016
Posts: 1
Location: United States
Concentration: Entrepreneurship, International Business
GRE 1: Q710 V430
WE: Information Technology (Computer Software)
Re: Formulas for Consecutive, Even, Odd Integers  [#permalink]

Show Tags

New post 03 Feb 2019, 17:28
Thanks bb for the inititaive.

Posted from my mobile device
Director
Director
User avatar
V
Joined: 06 Jan 2015
Posts: 587
Location: India
Concentration: Operations, Finance
GPA: 3.35
WE: Information Technology (Computer Software)
Premium Member
Re: Formulas for Consecutive, Even, Odd Integers  [#permalink]

Show Tags

New post 03 Feb 2019, 17:40
Sum of First "n" natural nos = n(n+1)/2
Sum of First "n" ODD natural nos = n^2
Sum of First "n" EVEN natural nos = n (n+1)
Sum of Square"n" natural nos = n(n+1)(2n+1)/6
_________________

आत्मनॊ मोक्षार्थम् जगद्धिताय च

Resource: GMATPrep RCs With Solution

GMAT Club Bot
Re: Formulas for Consecutive, Even, Odd Integers   [#permalink] 03 Feb 2019, 17:40
Display posts from previous: Sort by

Formulas for Consecutive, Even, Odd Integers

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


cron
Copyright

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

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®.