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

Question

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

Bunuel · Accepted Answer

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

Bunuel · Answer

2. Properties of Integers  
     Theory   
 Math: Number Theory 
 Number Properties: Tips and hints 
 Advanced Number Properties on GMAT (All 5 parts) 
 Do you make these 3 mistakes in GMAT Even-Odd Questions?  
 Everything about Factorials on the GMAT 
 Solving Complex Problems Using Number Line 
     Questions   
 Tough New Set: Number Properties Set 
 Power of a Number in a Factorial Problems 
 Number Properties Ability Quiz 
 Special Numbers and Sequences Problems 
 The E-GMAT Number Properties Knockout: 10 Great 700+ Ques for you! 
 Trailing Zeros Problems 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

abhi758 · Answer

Thanks Bunnel. Very well explained!

dixitraghav · Answer

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

vad3tha · Answer

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.

uday1409 · Answer

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

beattheheat · Answer

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

amitjathar · Answer

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.

pacifist85 · Answer

Hey,

Great, you reminded me of this formula!

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

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.

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

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.

stonecold · Answer

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

stonecold · Answer

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

KanishkM · Answer

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

BrentGMATPrepNow · Answer

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

hkhanexam · Answer

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

Basshead · Answer

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.

CEdward · Answer

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.

jack_tay · Answer

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

Bunuel · Answer

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 
     Theory   
 Inequalities Made Easy! 
 Inequalities: Tips and hints 
 Arithmetic with Inequalities 
 Wavy Line Method Application - Complex Algebraic Inequalities 
 Solving Quadratic Inequalities: Graphic Approach 
 Inequalities - Quadratic Inequalities 
 Graphic approach to problems with inequalities 
 Inequalities trick 
 INEQUATIONS(Inequalities) Part 1 
 INEQUATIONS(Inequalities) Part 2 
     Questions   
 Inequality and absolute value questions from my collection 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

Aditi.S · Answer

If S=18, which can be first 4 positive integers= (3+4+5+6).
If S=15, which can be first 5 positive integers= (1+2+3+4+5), then there will be multiple values of n, right? Or have I misinterpreted the "first n positive integers" ?

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

Prep Toolkit

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My Rewards