BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

Sum of odd numbers

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Master | Next Rank: 500 Posts
Posts: 429
Joined: Wed Sep 19, 2012 11:38 pm
Thanked: 6 times
Followed by:4 members

Sum of odd numbers

by [email protected] » Sun Jul 06, 2014 12:08 am
The sum of first N consecutive odd integers is N^2 . What is the sum of all odd integers between 13 and 39, inclusive?



351


364


410


424


450

My question is why can we find out the sum with the formula of evenly spaced set, I mean a set containing odd numbers has all evenly spaced integers the isn't (39+13)/2 giving us the mean which thereby helps us to find the sum of the set
Top
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Sun Jul 06, 2014 1:34 am
[email protected] wrote:The sum of first N consecutive odd integers is N^2 . What is the sum of all odd integers between 13 and 39, inclusive?



351


364


410


424


450
If we examine the sequence of consecutive odd integers, we get:
term1 = 1
term2 = 3
term3 = 5
term4 = 7
term5 = 9
term6 = 11
term7 = 13
.
.
.
term20 = 39

NOTICE that termk = 2k - 1


We want to evaluate 13 + 15 + 17 + . . . + 37 + 39
Notice that 13 + 15 + 17 + . . . + 37 + 39 = (1 + 3 + 5 + . . . + 37 + 39) - (1 + 3 + 5 + 7 + 9 + 11)
= (sum of first 20 odd integers) - (sum of first 6 odd integers)
= (20²) - (6²) [by applying the given formula]
= (400) - (36)
= 364

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Sun Jul 06, 2014 1:41 am
[email protected] wrote:The sum of first N consecutive odd integers is N^2 . What is the sum of all odd integers between 13 and 39, inclusive?



351


364


410


424


450

My question is why can we find out the sum with the formula of evenly spaced set, I mean a set containing odd numbers has all evenly spaced integers the isn't (39+13)/2 giving us the mean which thereby helps us to find the sum of the set
Sure, we can use the formula that says, If the terms are EQUALLY spaced, the sum = (average of first value and last value)(# of terms)

How many odd terms are there from 13 to 39 inclusive?
Well, judging from my first post, we're going from term7 to term20 inclusive.
So, the number of terms = 20 - 7 + 1 = 14

So, applying the formula, the sum = [(13 + 39)/2][14]
= [(52)/2][14]
= [26][14]
= 364

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
User avatar
Legendary Member
Posts: 1100
Joined: Sat May 10, 2014 11:34 pm
Location: New Delhi, India
Thanked: 205 times
Followed by:24 members

by GMATinsight » Sun Jul 06, 2014 6:08 am
The sum of first N consecutive odd integers is N^2 . What is the sum of all odd integers between 13 and 39, inclusive?



351


364


410


424


450

My question is why can we find out the sum with the formula of evenly spaced set, I mean a set containing odd numbers has all evenly spaced integers the isn't (39+13)/2 giving us the mean which thereby helps us to find the sum of the set
You can use it the way you have tried.

The average is certainly (13+39)/2 = 52/2 = 26

Total terms are = 20 (Odd numbers from 1 to 39) - 6 (Odd numbers from 1 to 12) = 14

Therefore required Sum = 26 x 14 = [spoiler]364 Option B[/spoiler]
"GMATinsight"Bhoopendra Singh & Sushma Jha
Most Comprehensive and Affordable Video Course 2000+ CONCEPT Videos and Video Solutions
Whatsapp/Mobile: +91-9999687183 l [email protected]
Contact for One-on-One FREE ONLINE DEMO Class Call/e-mail
Most Efficient and affordable One-On-One Private tutoring fee - US$40-50 per hour
Top
User avatar
Legendary Member
Posts: 1100
Joined: Sat May 10, 2014 11:34 pm
Location: New Delhi, India
Thanked: 205 times
Followed by:24 members

by GMATinsight » Sun Jul 06, 2014 6:11 am
Another way to look at it...

The Sum of first and last term (one pair) = (13+39) = 52

Total terms are = 20 (Odd numbers from 1 to 39) - 6 (Odd numbers from 1 to 12) = 14

Total pairs of terms = 14/2 = 7

Therefore sum of all such numbers = 7 x 52 = [spoiler]364 Option B[/spoiler]
"GMATinsight"Bhoopendra Singh & Sushma Jha
Most Comprehensive and Affordable Video Course 2000+ CONCEPT Videos and Video Solutions
Whatsapp/Mobile: +91-9999687183 l [email protected]
Contact for One-on-One FREE ONLINE DEMO Class Call/e-mail
Most Efficient and affordable One-On-One Private tutoring fee - US$40-50 per hour
Top
User avatar
Legendary Member
Posts: 1100
Joined: Sat May 10, 2014 11:34 pm
Location: New Delhi, India
Thanked: 205 times
Followed by:24 members

by GMATinsight » Sun Jul 06, 2014 6:15 am
But the Developer wanted this question to be solved in this way...
The sum of first N consecutive odd integers is N^2 . What is the sum of all odd integers between 13 and 39, inclusive?
Therefore,
Sum of first 20 Consecutive odd Integers (1 to 39) = 20^2 = 400
Sum of first 06 Consecutive odd Integers (1 to less than 13) = 6^2 = 36

Total numbers from 13 through 39 both inclusive = 400 - 36 = [spoiler]364 Option B[/spoiler]
"GMATinsight"Bhoopendra Singh & Sushma Jha
Most Comprehensive and Affordable Video Course 2000+ CONCEPT Videos and Video Solutions
Whatsapp/Mobile: +91-9999687183 l [email protected]
Contact for One-on-One FREE ONLINE DEMO Class Call/e-mail
Most Efficient and affordable One-On-One Private tutoring fee - US$40-50 per hour
Top
Post new topic Post Reply
• Page 1 of 1