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

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If S is the sum of the first n positive integers, what is [#permalink]  08 May 2010, 01:11
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If S is the sum of the first n positive integers, what is the value of n ?

(1) S < 20
(2) S^2 > 220
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Re: Inequalities [#permalink]  08 May 2010, 04:03
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abhi758 wrote:
If S is the sum of the first n positive integers, what is the value of n ?
(1) $$S < 20$$
(2) $$s^2 > 220$$

The sum of the first n positive integers $$S=\frac{n(n+1)}{2}$$.

(1) $$S < 20$$ --> $$\frac{n(n+1)}{2}< 20$$ --> $$n(n+1)<40$$ --> $$0<n<6$$ (n can 1, 2, 3, 4, 5). Not sufficient

(2) $$S^2 > 220$$ --> $$(\frac{n(n+1)}{2})^2> 220$$ --> $$n(n+1)>\sqrt{880}$$ --> $$\sqrt{880}$$ is slightly less than 30 --> $$n(n+1)>29$$ --> $$n>4$$ (n can be 5, 6, 7, ...). Not sufficient.

(1)+(2) Intersection of values of n from (1) and (2) is n=5. Sufficient.

OR, just write down several values of S. S= 1, 3, 6, 10, 15, 21, 28, ...

(1) $$S < 20$$. S=1, 3, 6, 10, 15. Not sufficient

(2) $$S^2 > 220$$. S=15, 21, ... Not sufficient.

(1)+(2) Intersection of values of n from (1) and (2) is S=15 --> n=5. Sufficient.

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Re: Inequalities [#permalink]  08 May 2010, 11:28
Thanks Bunnel. Very well explained!
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Re: Inequalities [#permalink]  08 May 2010, 23:32
somehow I have less problem with the math and more with the sufficient /not sufficient concept. But thanks. The more problems I look at the more it makes sense. (still easier to see that with the explanation)
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Re: Inequalities [#permalink]  16 May 2010, 12:08
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easy one this time
s<20
so N could be 2 to 5 (N greater than 5 will exceed the limit s<20) this condition alone not sufficient.
statement B - S^2 > 220 N clould be 5,6,7,8 anything equal or greater than 5 hence not sufficient.
combining the two , we get N=5 which is the answer
hence C
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Re: If S is the sum of the first n positive integers, what is [#permalink]  06 Feb 2014, 02:28
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Re: If S is the sum of the first n positive integers, what is [#permalink]  29 Jul 2014, 14:24
abhi758 wrote:
If S is the sum of the first n positive integers, what is the value of n ?

(1) S < 20
(2) S^2 > 220

First n positive intergers: 1, 2, 3, 4, 5, 6, 7.
Sum of them could be: 3,6,10,15,21

(1) S < 20 -> n could be 1, 2, 3, 4, 5, -> insufficient
(2) S^2 > 220
220 = 11*2*10 = 11*4*5 -> S > square root of 220 = 2 * sqrt(55)
7^2 < 55 < 8^2 -> S > (about) 14

--> C ( S =15)

That is everything on my note in 1 min 20 sec.
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Re: If S is the sum of the first n positive integers, what is   [#permalink] 29 Jul 2014, 14:24
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