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

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.

Answer: C.

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.

Answer: C.
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2. Properties of Integers

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For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Thanks Bunnel. Very well explained!

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

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.

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

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

Answer C

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.

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.

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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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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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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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

Answer: C

Cheers,
Brent

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

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

(1) S < 20
n(n+1)/2 < 20
n(n+1) <40
Now, try out some numbers:
1(1+1)=2 < 40
2(2+1)=6 < 40
3(3+1)=12 < 40
4(4+1)=20 < 40
5(5+1)=30 < 40
6(6+1)=42 < 40 (X) Therefore all the positive integers less than 5 can be the answer.
Since we can't define the only value for n, NOT SUFFICIENT

(2) S^2 > 220
S > √220
S > 15
n(n+1)/2 > 15 = n(n+1) > 30
Trying the same strategy to test out the numbers, you find that n could be any number greater than 5. For example:
5(5+1)=30 > 30
6(6+1)=35 > 30
Therefore all the positive integers greater than 5 can be the answer.
Since we can't define the only value for n, NOT SUFFICIENT

When you combine both (1) and (2), you see that n = 5

Answer: C

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

(1) S < 20

If S is less than 20:

1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15

There are a few possibilities for n; insufficient.

(2) S^2 > 220

15^2 = 225
This means S is greater than 14. S could be 1,000 or 10,000 -- insufficient.

(1&2) The only sum that is greater than 14 but less than 20 is 15. This means n is 5. Sufficient.

Answer is C.
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Question regarding "First n positive integers",

These integers have to be from 1 onwards i.e distinct or can these be 1,1,2,3,3 etc ?

Almost pressed E.

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

(1) S < 20
This sum can be anything less than 20...lots of possibilities
1 + 2 =3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10...
1 + 2 + 3 + 4 + 5 = 15
Insufficient

(2) S^2 > 220
There's no upper limit to the sum so S can be anything ≥ 15

15^2 = 225
21^2 = some number

C: S = 15 so 1 + 2 + 3 + 4 + 5 ...n = 5
n cannot be 6 because ....
1 + 2 + 3 + 4 + 5 + 6 = 21 <--- violates statement 1.

C.
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