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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]

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22 Jul 2011, 19:16
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-s-is-the- ... 93852.html
[Reveal] Spoiler: OA
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Joined: 21 Apr 2011
Posts: 3
Re: If S is the sum of the first n positive integers, what is [#permalink]

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22 Jul 2011, 23:01
To find the sum of evenly spaced set of finite numbers, you can use the arithmetic progression formula: Sn = (n/2)[A1 + (n-1)d], where, n = the number of terms
A1= the first term
d = the common difference
In the question, we're asked for n.

Stat. 1: S<20
(n/2)[A1 + (n-1)d] <20 solving for n shall give you: note that d=1
n <\sqrt{40}
This means n can be 6,5,4,3,2 or 1. So insufficient.
Stat. 2: S^2 >220
S> \sqrt{220}
So S >14 Here note that \sqrt{220}is between 14 and 15.
Therefore, using the previous formula
n > \sqrt{28}
This means n can be 6,7,8,9,10....So insufficient
Combining statement 1 and statement 2 leaves us with only number 6 as a valid value for n; therefore, it is sufficient.
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Re: If S is the sum of the first n positive integers, what is [#permalink]

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23 Jul 2011, 00:23
1
KUDOS
tracyyahoo 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

Sum of first n positive integer S = n * (n+1) / 2

Stmt1: S < 20. For n=5 S = 5*6/2 =15
For n=4 S=4*5/2 = 10
Hence n can be 5 or 4 or 3.... Insufficient.

Stmt2: S^2 >220
For n=5, S=15 and S^2 = 225 > 220
For n=6, S=21 and S^2 = 441 > 220
Again, two values. Insufficient.

Combining, From 1, n can be 5,4,3... From 2, n can be 5,6,7.... Common value is 5. hence sufficient.

OA C.
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Re: If S is the sum of the first n positive integers, what is [#permalink]

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23 Jul 2011, 03:56
S is the sum of n integer, not the averge. Big brother....Stmt1: S < 20. For n=5 S = 5*6/2 =15?

tracyyahoo 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

Sum of first n positive integer S = n * (n+1) / 2

Stmt1: S < 20. For n=5 S = 5*6/2 =15
For n=4 S=4*5/2 = 10
Hence n can be 5 or 4 or 3.... Insufficient.

Stmt2: S^2 >220
For n=5, S=15 and S^2 = 225 > 220
For n=6, S=21 and S^2 = 441 > 220
Again, two values. Insufficient.

Combining, From 1, n can be 5,4,3... From 2, n can be 5,6,7.... Common value is 5. hence sufficient.

OA C.
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Re: If S is the sum of the first n positive integers, what is [#permalink]

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23 Jul 2011, 03:58
Too complicated, hard to understand~~

yefetarisira wrote:
To find the sum of evenly spaced set of finite numbers, you can use the arithmetic progression formula: Sn = (n/2)[A1 + (n-1)d], where, n = the number of terms
A1= the first term
d = the common difference
In the question, we're asked for n.

Stat. 1: S<20
(n/2)[A1 + (n-1)d] <20 solving for n shall give you: note that d=1
n <\sqrt{40}
This means n can be 6,5,4,3,2 or 1. So insufficient.
Stat. 2: S^2 >220
S> \sqrt{220}
So S >14 Here note that \sqrt{220}is between 14 and 15.
Therefore, using the previous formula
n > \sqrt{28}
This means n can be 6,7,8,9,10....So insufficient
Combining statement 1 and statement 2 leaves us with only number 6 as a valid value for n; therefore, it is sufficient.
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Posts: 1945
Re: If S is the sum of the first n positive integers, what is [#permalink]

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23 Jul 2011, 07:01
tracyyahoo 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

Pls tell me why? THANK U~~

Sum of n natural number = n(n+1)/2

1)
n(n+1)/2<20
n(n+1)<40
{1, 2}=2
{2, 3}=6
{3, 4}=12
{4, 5}=20
{5, 6}=30
{6, 7}=42>40. Ignore & Stop.

1<=n<=5
Not Sufficient.

2)

S^2 >220
Square root both sides:
S >14
n(n+1)/2>14
n(n+1)>28
{5,6}=30
{6,7}=42
.
.
.
n>=5
Not Sufficient.

Together:
n=5

Ans: "C"
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Posts: 263
Re: If S is the sum of the first n positive integers, what is [#permalink]

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23 Jul 2011, 13:09

1) S < 20.

1+2+3+4+5+6 = 21. 21 is greater than 20. Therefore, statement 1 tells us that 1<=n<=5

2) S^2 > 220. S > 15.

Statement 2 tells us that the sum is greater than or equal to 15. This also tells us that n has to be greater than or equal to 5.

Combined: We get statement 1 which states that n is less than or equal to 5 and statement 2 which states n is greater than or equal to 5. Therefore, N is equal to 5. Hence Sufficient from both statements 1 and 2.
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Re: If S is the sum of the first n positive integers, what is [#permalink]

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27 Nov 2017, 00:35
If S is the sum of the first n positive integers, what is the value of n ?

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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-s-is-the- ... 93852.html
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Re: If S is the sum of the first n positive integers, what is   [#permalink] 27 Nov 2017, 00:35
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