Data Sufficiency

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

1) S < 20
2) S² > 220

Target question: What is the value of n?

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

Statement 1: S < 20
There are several values of n that satisfy this condition. Here are two:
Case a: n = 2. Here, S = 1 + 2 = 3, which is less than 20
Case b: n = 3. Here, S = 1 + 2 + 3 = 5, which is less than 20
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: S² > 220
There are several values of n that satisfy this condition. Here are two:
Case a: n = 50. Here, S = 1 + 2 + .... + 50, so S² = is definitely bigger than 220.
Case b: n = 51. Here, S = 1 + 2 + .... + 51, so S² = is definitely bigger than 220.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
USEFUL FORMULA: For any positive integer n, the sum of the first n positive integers equals n(n+1)/2
Let's examine a few possible values of n and the subsequent values of S and S²

n = 4. Applying the formula, we see that S = 4(4+1)/2 = 10
So, S = 10 and S² = 100
This meets the statement 1 condition that S < 20, but it does NOT meet the statement 2 condition that S² > 220
So, n cannot equal 4.

n = 5. Applying the formula, we see that S = 5(5+1)/2 = 15
So, S = 15 and S² = 225
This meets the statement 1 condition that S < 20, AND it meets the statement 2 condition that S² > 220
So, n COULD EQUAL 5.

n = 6. Applying the formula, we see that S = 6(5+1)/2 = 21
So, S = 21 and S² = 441
This does NOT meet the statement 1 condition that S < 20, but it meets the statement 2 condition that S² > 220
So, n CANNOT equal 6.

As you can see, any value of n greater than 5 will not meet the statement 1 condition.
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

RiyaR 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

This is my understanding of the problem:
Statement A on its own doesnt tell us the value of S as there are various possibilities. Statement B too doesnt provide a definitve answer. Now if we combine both the staments, we know that S is less than 20 and S^2 is greater than 220 so S can be 15(15^2=225),16(16^2=256),17(17^2=289). So according to me, the answer should be e.But I was told the answer is c. Help!

Hi RiyaR,

The target question asks us to find the value of n. As you can see from my solution, there is only one possible value of n.
I believe that you are answering a DIFFERENT target question. You are trying to determine the value of S. In so doing, you are neglecting the fact that S is the sum of the first n positive integers

Cheers,
Brent

Hi RiyaR,

Brent has made an important point about how you were dealing with the value of S. This type of error usually occurs when someone isn't taking enough notes. Here's how you can avoid making this type of mistake - quickly jot down a few examples (for reference).

S = sum of the first N POSITIVE integers

If....
N = 1, then S = 1
N = 2, then S = 1 + 2 = 3
N = 3, then S = 1 + 2 + 3 = 6
N = 4, then S = 1 + 2 + 3 + 4 = 10
N = 5, then S = 1 + 2 + 3 + 4 + 5 = 15

N = 6, then S = 1 + 2 + 3 + 4 + 5 + 6 = 21

From this, we can see that S CANNOT be "just any value"; it can only be certain values. This is one of the ways that the GMAT tests your attention-to-detail. The math isn't difficult, in and of itself, but these tasks place a premium on thorough, organized work, so you have to be ready to PROVE that your assumptions are correct.

GMAT assassins aren't born, they're made,
Rich