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# The Sum of first N consecutive odd integers is N^2. What is

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The Sum of first N consecutive odd integers is N^2. What is [#permalink]  02 Aug 2006, 16:34
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The Sum of first N consecutive odd integers is N^2. What is the sum of all odd integers between 13 and 39 inclusive.

A. 351
B. 364
C. 410
D. 424
E. 450
[Reveal] Spoiler: OA
Manager
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B....

1- 11 are the first 6 odd intergers; sum = 6^2 = 36

1- 39 are the first 20 odd intergers; sum = 400

sum of odd nos frm 13- 39 = 400-36 = 364
Manager
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Use AP and you woudn't need the supplied formula:

a(1) = 13; a(n) = a(1) + (n-1)d = 39 (where d =2).
Sum = n/2[a(1) + a(n)] = 364, where n = 14
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Number of odd numbers from 1-39 = 39+1/2 = 20
Sum = 400

Number of odd numbers between = 1-12 = (12 - 1 +1)/2 = 6
sum = 36

Sum between 13-39 inclusive = 400 - 36 = 364
Manager
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I also used AP without using N^2 condition. For me, sometimes, it is easeir to just stick to basics and do the calcs. This way, at least, I know I will arrive at the correct answer sooner or later...
Director
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I would suggest using the formula, for the simple reason that instead of an AP the problem might refer to some fancy formula for an unknown sequence of numbers.

Anyway here we go...

For odd integers till 39, number of integers = 20
Sum = 20^2 = 400
For odd integers till (but not including 13), number of integers = 6
Sum = 6^2 = 36
Difference = 364 = B
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Re: PS : Number Prop. [#permalink]  04 Aug 2006, 05:43
ghantark 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

OA later

i never knew this formulea that sum of n consecutive odd integers = n^2.
however, we can use regular formula:
the sum of consecutive odd integers from 13-39 = n (L-s)/2 = [{(39-13)/2} +1][39-13] = 14 x 26 = 364
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Re: PS : Number Prop. [#permalink]  05 Aug 2006, 00:03
Professor wrote:
ghantark 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

OA later

i never knew this formulea that sum of n consecutive odd integers = n^2.
however, we can use regular formula:
the sum of consecutive odd integers from 13-39 = n (L-s)/2 = [{(39-13)/2} +1][39-13] = 14 x 26 = 364

It can be derived from the regular formula for A.P.

1+3+5+....=n/2[2.1+(n-1).2] = n^^2 !!
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Re: PS : Number Prop. [#permalink]  05 Aug 2006, 14:37
ghantark 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

OA later

Sum of odd integers from 1 to 39 inclusive = 20^2=400
sum of odd integers from 1 to 11 inclusive = 6^2 = 36
therfore sum of odd integers between 13 and 39 incl = 400 -36 = 364

Hence B

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Re: The Sum of first N consecutive odd integers is N^2. What is [#permalink]  07 Jun 2012, 00:58
I method -
the sum of odd numbers till 39(inclusive) =n^2=20^2

the sum of odd numbers till 11 (inclusive) =n^2=6^2

20^2 -6^2=(20-6)(20+6)=26*14=364

================

II method
n* (a1+an)/2
n=((39-13)/2)+1=14

(39+13)/2* 14=26*14=364
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Re: The Sum of first N consecutive odd integers is N^2. What is [#permalink]  21 Jan 2015, 17:44
ghantark 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.

A. 351
B. 364
C. 410
D. 424
E. 450

from the question I see that it ask about the sum of odd numbers between 13 and 39 not from 13 to 39 therefore, 13 and 39 are not includes in the sum.So, there are 20 odd numbers from 1 to 39 so 20^2 = 400 and from 1 to 13 there are 7 odd numbers so it will be 7^2=49
400-49=351

if I am no right what will be the answer if the question was ask about the sum of odd numbers from 13 to 39
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Re: The Sum of first N consecutive odd integers is N^2. What is [#permalink]  21 Jan 2015, 18:59
Expert's post
Hi All,

These types of "sum of a sequence" questions can be approached in a number of different ways - you can use various formulas or you can approach this prompt by pattern-matching and minimizing the math that you have to do. For this question, you can use "bunching"....

We're dealing with a sequence of CONSECUTIVE ODD INTEGERS: 13 to 39, inclusive. We're asked for the SUM of this group.

1) Start with the sum of the smallest and the biggest: 13 + 39 = 52
2) Now look at the 'next smallest' and the 'next biggest': 15 + 37 = 52

From this, you can see that you're just going to be adding up a bunch of 52s. We DO have to check to see if there's a "middle" term in this sequence that doesn't get "bunched" though. To determine if that middle term exists, we just have find the last few 52s in the group....

21 and 31
23 and 29
25 and 27

Now we have proof that there is no middle term. We have 7 bunches of 52.

7(52) = 364

[Reveal] Spoiler:
B

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Re: The Sum of first N consecutive odd integers is N^2. What is [#permalink]  22 Jan 2015, 02:13
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23a2012 wrote:
ghantark 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.

A. 351
B. 364
C. 410
D. 424
E. 450

from the question I see that it ask about the sum of odd numbers between 13 and 39 not from 13 to 39 therefore, 13 and 39 are not includes in the sum.So, there are 20 odd numbers from 1 to 39 so 20^2 = 400 and from 1 to 13 there are 7 odd numbers so it will be 7^2=49
400-49=351

if I am no right what will be the answer if the question was ask about the sum of odd numbers from 13 to 39

Its asking sum of odd integers between 13 and 39 inclusive, so we require to minus sum of odd integers up to 11 inclusive

$$= 20^2 - 6^2 = 26 * 14 = 364$$

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Re: The Sum of first N consecutive odd integers is N^2. What is [#permalink]  22 Jan 2015, 15:23
ghantark 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.

A. 351
B. 364
C. 410
D. 424
E. 450

Thank you for EMPOWERgmatRichC and PareshGmat

But I still do not understand what will be the answer if the question was ask about the sum of odd numbers from 13 to

39 and why we do not includ 13?
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Re: The Sum of first N consecutive odd integers is N^2. What is [#permalink]  22 Jan 2015, 16:37
Expert's post
Hi 23a2012,

My explanation (and everyone else's too) includes 13 and 39 in the sum. The prompt uses the phrase "ALL odd integers between 13 and 39, INCLUSIVE." This means that we have to INCLUDE 13 and 39 and all of the explanations do that.

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Re: The Sum of first N consecutive odd integers is N^2. What is   [#permalink] 22 Jan 2015, 16:37
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