Hi All,
We're told that in any sequence of N NON-ZERO numbers, a pair of consecutive terms with opposite signs represents a sign change (for example, the sequence -2, 3, -4, 5 has three sign changes). We're asked if the sequence of nonzero numbers s1, s2, s3, . . . , sn has an EVEN number of sign changes. This is a YES/NO question. This question can be solved with a bit of Arithmetic and TESTing VALUES.
(1) sk = (-1)^k for all positive integers k from 1 to N.
While the information in Fact 1 might look complex, it actually defines how the sequence 'works.' The initial terms - and the pattern behind the sequence - are...
1st term = (-1)^1 = -1
2nd term = (-1)^2 = +1
3rd term = (-1)^3 = -1
4th term = (-1)^4 = +1
Etc.
We know how the sequence repeats, but we DON'T know how many terms are in it, so we don't know exactly how many sign changes there are (and thus, we don't know if the total number of changes is even OR odd).
IF....
There are 3 terms, then we have -1, +1, -1 and two sign changes - so the answer to the question is YES.
There are 4 terms, then we have -1, +1, -1, +1 and three sign changes - so the answer to the question is NO.
Fact 1 is INSUFFICIENT
(2) N is ODD.
Fact 2 tells us the number of terms in the sequence, but we know nothing about any of their values, so we don't know how many sign changes occur.
Fact 2 is INSUFFICIENT
Combined, we know...
sk = (-1)^k for all positive integers k from 1 to N.
N is ODD.
We know exactly how the sequence 'works' and that there are an ODD number of terms.
IF
There are 3 terms, then we have -1, +1, -1 and two sign changes - so the answer to the question is YES.
There are 5 terms, then we have -1, +1, -1, +1, -1 and four sign changes - so the answer to the question is YES.
There are 7 terms, then we have -1, +1, -1, +1, -1, +1, -1 and six sign changes - so the answer to the question is YES.
Etc.
Whenever there are an odd number of terms, the answer to the question is YES.
Combined, SUFFICIENT
Final Answer:
GMAT assassins aren't born, they're made,
Rich