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

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If S is the sum of the first n positive integers, what is the value of  [#permalink]

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08 May 2010, 02:11
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15
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Difficulty:

55% (hard)

Question Stats:

65% (01:54) correct 35% (01:52) wrong based on 386 sessions

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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

DS84302.01
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Joined: 02 Sep 2009
Posts: 60647
Re: If S is the sum of the first n positive integers, what is the value of  [#permalink]

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08 May 2010, 05:03
5
4
abhi758 wrote:
If S is the sum of the first n positive integers, what is the value of n ?
(1) $$S < 20$$
(2) $$s^2 > 220$$

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.

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

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08 May 2010, 12:28
Thanks Bunnel. Very well explained!
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Re: If S is the sum of the first n positive integers, what is the value of  [#permalink]

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16 May 2010, 13:08
1
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 the value of  [#permalink]

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29 Jul 2014, 15:24
1
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 the value of  [#permalink]

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18 Dec 2014, 06:55
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Re: If S is the sum of the first n positive integers, what is the value of  [#permalink]

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18 Dec 2014, 07:53
2
1.S<20
(n(n+1)/2)<20
for n = 1 to 5 the above equation is valid. so we cant say the exact valeu of n . Insufficient

2. (n^2(n+1)^2)/4>220
for n>4 the above equation is valid. so we cant say the exact valeu of n . Insufficient

1+2>> there is only one value that satisfies both the equations i.e n=5
from 1, 1<=n<=5 and from 2, n>4
common value is 5. Sufficinet

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

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19 Dec 2014, 10:51
1
If S is the sum of the first n positive integers, what is the value of n ?

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

Solution:
Sum of first positive n numbers is defined as S= n*(n+1)/2

Statement 1: S < 20
Five distinct values of n satisfies this condition (n={1,2,3,4,5}) - Insufficient

Statement 2: S^2 > 220
Since S is sum of positive integers (ignoring negative values), S > 14.
Again multiple values of n satisfies this condition (n>=5) - Insufficient

Combining Statement 1 and 2: Unique solution n = 5

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

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20 Dec 2014, 17:08
1
S = sum of first positive integers = n(n+1)/2

(1) S < 20
if the number of integers are 4, S = 4*5/2 = 10
if the number of integers are 5, S = 5*6/2 = 15. Insufficient
(2) S^2 > 220
S^2 goes till infinite. Insufficient.

If considered, 1 and 2, then S = 15. Sufficient.

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

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08 Feb 2015, 06:11
Hey,

Great, you reminded me of this formula!

However, in this case, we don't really need it.

[1] S<20
This means that the numbers are: 1,2,3,4,5 = 15. NS, because there is a range of possible sums using these values.

[2] s^2>220
This is not sufficient on its own, again becase there is a range of possible sums.

[1,2] Sufficient.
15^2 = 225 (1+2+3+4+5).
10^2 = 100 (1+2+3+4)

So, only 15^2 = 225 can be the case.
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Re: If S is the sum of the first n positive integers, what is the value of  [#permalink]

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11 Dec 2016, 11:26
Excellent Question.
Here is what i did in this Question -->

From 1 =>
S=1,3,6,10,15
Not sufficient
From 2=>
N can be 1000 or 10000 or anything basically with s^2>220 restriction.

Hence insufficient

Combing the two statements S must be 15
Hence n must be 5

Hence C

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

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16 Dec 2016, 18:34
1
This is a great Official Question.
Here is what i did in this one =>
WE need the value of n

Statement 1-->
S<20

n=1=>S=1
n=2=>S=3
n=3=>S=6
n=4=>S=10
n=5=>S=15

Hence n can be {1,2,3,4,5}
Not sufficient

Statement 2-->
Here n can take infinite values.
E.g=> n=1000902439380
n=1489312843
etc.
Not sufficient

Combining the two statements-->
S=15 is the only value that would satisfy both the conditions

Hence S=15
n=5

Hence sufficient

Hence C

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

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07 Feb 2019, 03:55
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

n can be 3, 4, 5

From 1, sum can be 10,15(for n =4 and n=5)

From 2, n can be any value after 5,6,7,8

Combination will give only one value for n

n = 5, sum = 15, 225> 220

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

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15 Dec 2019, 07:52
Top Contributor
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

Given: S is the sum of the first n positive integers

Target question: What is the value of n ?

Statement 1: S < 20
If n = 5, then S = 1 + 2 + 3 + 4 + 5 = 15
If n = 6, then S = 1 + 2 + 3 + 4 + 5 + 6 = 21
Since S < 20, we know that n < 6
This means n COULD equal 1, 2, 3, 4, or 5
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: S² > 220
Let's test some values.
If n = 4, then S = 1 + 2 + 3 + 4 = 10, which means S² = 10² = 100
If n = 5, then S = 1 + 2 + 3 + 4 + 5 = 15, which means S² = 15² = 225
Since S² > 220, we know that n > 4
This means n COULD equal 5 or 6 or 7 or 8 or .....
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that n < 6
Statement 2 tells us that n > 4
Combine the inequalities to get: 4 < n < 6
So, it MUST be the case that n = 5
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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Re: If S is the sum of the first n positive integers, what is the value of   [#permalink] 15 Dec 2019, 07:52
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