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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) \(s^2 > 220\)

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.

somehow I have less problem with the math and more with the sufficient /not sufficient concept. But thanks. The more problems I look at the more it makes sense. (still easier to see that with the explanation)

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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