Sum of odd numbers

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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

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Source: Beat The GMAT — Problem Solving | Sun Jul 06, 2014 12:08 am

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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)
= (20Â²) - (6Â²) [by applying the given formula]
= (400) - (36)
= 364

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; 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

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

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]

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

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]

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

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]

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