The next number in a certain sequence is defined by

[AAPL]

Manhattan Prep

The next number in a certain sequence is defined by multiplying the previous term by some positive constant \(k\), where \(k \neq 1\). How many of the first nine terms in this sequence are greater than 1?

1) The ninth term in this sequence is 81.
2) the fifth term in this sequence is 1.

OA B

[Ian Stewart]

If our sequence contains positive values, and if we're multiplying by some positive constant (different from one) to produce each subsequent term, the sequence is either constantly increasing (if the constant is greater than 1) or constantly decreasing. (if the constant is less than 1). Either way, if the 5th term, which is in the middle of our nine terms, is equal to 1, then four terms will be greater than 1 and four will be less than 1 -- either the 1st through 4th terms are greater than 1 if the sequence is decreasing, or the 6th through 9th are if it is increasing.

Statement 1 isn't very useful -- if the constant is 1/2, say, then all of the terms can be greater than 1, but if the constant is 1,000,000,000, then only one of the terms is, so the answer is B.