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

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Re: If S is the sum of the first n positive integers, what is the value of [#permalink]
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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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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abhi758
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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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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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
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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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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]
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]
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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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]
abhi758
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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abhi758
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

Answer: C

Cheers,
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Re: If S is the sum of the first n positive integers, what is the value of [#permalink]
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
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Re: If S is the sum of the first n positive integers, what is the value of [#permalink]
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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Re: If S is the sum of the first n positive integers, what is the value of [#permalink]
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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Re: If S is the sum of the first n positive integers, what is the value of [#permalink]
Bunuel
abhi758

$$n(n+1)<40$$ --> $$0<n<6$$ (n can 1, 2, 3, 4, 5).

Hi, how do you solve for n(n+1)<40 ?
Thanks
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Re: If S is the sum of the first n positive integers, what is the value of [#permalink]
Expert Reply
jack_tay
Bunuel
abhi758

$$n(n+1)<40$$ --> $$0<n<6$$ (n can 1, 2, 3, 4, 5).

Hi, how do you solve for n(n+1)<40 ?
Thanks

Since n is a positive integer, the most efficient way to solve this is by plugging in numbers. Check when the product of two consecutive integers is less than 40. If n is 6, then n(n + 1) equals 42, which is greater than 40. However, if n is 5, then n(n + 1) equals 30, which is less than 40. Thus, n can be any positive integer less than or equal to 5, so n can be 1, 2, 3, 4, or 5.

To solve other types of inequalities, check the links below.

9. Inequalities

For more check Ultimate GMAT Quantitative Megathread

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