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# If the sum of the first n positive integers is S, what is

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Manager
Joined: 04 Feb 2011
Posts: 52
Location: US
If the sum of the first n positive integers is S, what is  [#permalink]

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08 Mar 2011, 05:07
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If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S ?

(A) S/2
(B) S
(C) 2S
(D) 2S + 2
(E) 4S
Math Expert
Joined: 02 Sep 2009
Posts: 52294
Re: p/s sum of integers  [#permalink]

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08 Mar 2011, 05:32
6
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Lolaergasheva wrote:
If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S ?
(A) S/2
(B) S
(C) 2S
(D) 2S + 2
(E) 4S

1+2+3+...+n=S
2+4+6+...+2n=2(1+2+3+...+n)=2S.

Or you can simply take n=2 --> 1+2=3=S --> 2+4=6=2S.

Generally:
Sum of n first integers: $$1+2+...+n=\frac{1+n}{2}*n$$

Sum of n first 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: $$a_n=2n-1$$. Given $$n=5$$ first odd 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: $$a_n=2n$$. Given $$n=4$$ first positive even integers, then their sum equals to $$2+4+6+8=4(4+1)=20$$.

For more check Number Theory chapter of Math Book: math-number-theory-88376.html
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Director
Status: Impossible is not a fact. It's an opinion. It's a dare. Impossible is nothing.
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Re: p/s sum of integers  [#permalink]

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08 Mar 2011, 05:24
1
2
Let n = 6.
S = n(n+1)/2 = 3 * 7 = 21
Sum of first 6 even = 2 + 4 + 6 + 8 + 10 + 12 = 42 = 2S

Hence C.

Lolaergasheva wrote:
If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S ?
(A) S/2
(B) S
(C) 2S
(D) 2S + 2
(E) 4S
Director
Joined: 01 Feb 2011
Posts: 658
Re: p/s sum of integers  [#permalink]

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09 Mar 2011, 16:22

Given sum of 1 2 3.....n = S

sum of 2 , 4 , 6 ....2n = 2[sum(1 2 3....n) ] = 2S
Senior Manager
Joined: 08 Nov 2010
Posts: 336
Re: p/s sum of integers  [#permalink]

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12 Mar 2011, 13:18
hmm... lets say the series is

{1,2,3,4,5} then S = 15
the sum of the even numbers - 6.

what am i missing?
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Math Expert
Joined: 02 Sep 2009
Posts: 52294
Re: p/s sum of integers  [#permalink]

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12 Mar 2011, 13:30
1
144144 wrote:
hmm... lets say the series is

{1,2,3,4,5} then S = 15
the sum of the even numbers - 6.

what am i missing?

The sum of the first 5 positive integers is 1+2+3+4+5=15;
The sum of the first 5 positive even integers is 2+4+6+8+10=30=2*15 (so the second sum shouldn't be the sum of the even numbers from the first list as you did).
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Senior Manager
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Re: p/s sum of integers  [#permalink]

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12 Mar 2011, 13:53
damn... got confused bc of the S.

thanks!
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Re: p/s sum of integers  [#permalink]

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12 Mar 2011, 19:43
4
1
Lolaergasheva wrote:
If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S ?
(A) S/2
(B) S
(C) 2S
(D) 2S + 2
(E) 4S

I see that most people would solve it by taking some value of n... that's great... One suggestion - don't shy away from taking the lowest and simplest value.
I read 'sum of the first n positive integers', I say 'n = 1'... First 1 positive integer is 1 so S = 1. First 1 positive even integer will be 2 which is equal to twice of S. Only one options works out here so you are done. (In some cases, more than one option could work out e.g. if $$2S^2$$ were another option. In that case you might want to check for n = 2 as well)
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Re: If the sum of the first n positive integers is S, what is  [#permalink]

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09 Jan 2017, 19:02
Intern
Joined: 14 Oct 2016
Posts: 32
Location: India
WE: Sales (Energy and Utilities)
Re: If the sum of the first n positive integers is S, what is  [#permalink]

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08 Sep 2017, 19:52
Well, this one can be solved in less than 10 secs if we know some formulas

Sum of firs n natural numbers = n(n+1)/2

Sum of first n even natural numbers = n(n+1)

Sum of first n off natural numbers is n^2

ok
we are given that n(n+1)/2=S
that means n(n+1)=2S and we know that n(n+1) is sum of first n even natural numbers . And that's the answer.
2S
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Abhimanyu

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Joined: 12 Sep 2017
Posts: 53
Re: If the sum of the first n positive integers is S, what is  [#permalink]

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31 Dec 2018, 19:02
AbhimanyuDhar wrote:
Well, this one can be solved in less than 10 secs if we know some formulas

Sum of firs n natural numbers = n(n+1)/2

Sum of first n even natural numbers = n(n+1)

Sum of first n off natural numbers is n^2

ok
we are given that n(n+1)/2=S
that means n(n+1)=2S and we know that n(n+1) is sum of first n even natural numbers . And that's the answer.
2S

Hello!

Could someone mind to explain to me where do we infer that n(n+1) must be divided by 2?

I am really confused about that.

Thank you so much!
Re: If the sum of the first n positive integers is S, what is &nbs [#permalink] 31 Dec 2018, 19:02
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