Data Sufficiency

Hi Needgmat,

This DS question tells us that each term in a sequence is equal to the PREVIOUS term multiplied by a POSITIVE CONSTANT.

For example, the sequence 1, 2, 4, 8, 16 would fit this definition (another example would be 16, 8, 4, 2, 1, 1/2, etc.). We don't know any of the terms though and we don't know the constant (it could be either an integer, fraction or mixed number). We DO know that since we're multiplying by a positive constant that the sequence of numbers either "increases" or "decreases."

We're asked how many of the first 9 terms are greater than 1? It's interesting that the question asks how many are greater than 1.

Fact 1: the 9th term = 81

IF k = 3, then the terms (working backwards from the 9th term....) are:
81, 27, 9, 3, 1, 1/3, 1/9, 1/27, 1/81
Here, the number of terms greater than 1 = 4

IF k = 1/3, then the terms (working backwards from the 9th terms....) are:
81, 243, 729, then they get bigger and bigger.....
Here, the number of terms greater than 1 = 9
Fact 1 is INSUFFICIENT

Fact 2: the 5th term = 1

Since the sequence either increases or decreases, we'd have...
4 numbers less than 1, 1, 4 numbers greater than 1
or
4 numbers greater than 1, 1, 4 numbers less than 1

Regardless of which option, we end up with exactly 4 terms.
Fact 2 is SUFFICIENT

Final Answer: B

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

Needgmat wrote:The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 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.

OAB

Please explain.

S1: We are given that T9 = 81 => T8 = 81/k, T7 = 81/k^2, ... T1 = 81/k^8. But we do not the value of k. Insufficient!

S2: We are given that T5 = 1 => T6 = k, T7 = k^2, ... T10 = k^5, .... But we do not the value of k. However, it is not insufficient.

If k > 1, then terms T6, T7, T8, and T9 are greater than 1. The answer is four.

If k < 1, then terms T1, T2, T3, and T4 are greater than 1. The answer is still four. We do not need to know the value of k. Sufficient!

We cannot reach a conclusion applying the same logic for S1. If k < 1, we get to know that there would be nine terms greater than 1. However, if k > 1, we cannot reach to a term lying between T1 and T9, that is less than 1.

Hope this helps!

-Jay

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Needgmat wrote:The next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 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.

We are given that the next number in a certain sequence is defined by multiplying the previous term by some positive constant k, where k ≠ 1, and we need to determine how many of the first nine terms in this sequence are greater than 1.

Statement One Alone:

The ninth term in this sequence is 81.

Without knowing any other terms in the sequence or the value of the positive constant k we are multiplying, we can't determine the number of terms in the sequence that are greater than 1. For example, if k = 3, then by going backward we have:

9th term = 81, 8th term = 27, 7th term = 9, 6th term = 3, 5th term = 1, 4th term = 1/3, and so on.

In this case, we have 4 terms that are greater than 1.

However, if k = 9, then by going backward again we have:

9th term = 81, 8th term = 9, 7th term = 1, 6th term = 1/9, and so on.

In this case we have only 2 terms that are greater than 1. Therefore, statement one alone is not sufficient. We can eliminate answer choices A and D.

Statement Two Alone:

The fifth term in this sequence is 1.

Since there are 9 total terms, we see that the 5th term is the middle term in our sequence. In other words there are 4 terms below the 5th term and 4 terms above the 5th term. Since we know that k is positive, it's either a positive proper fraction or a number greater than 1. Thus, regardless of whether k is a positive proper fraction or a number greater than 1, there will be 4 numbers above 1 and 4 numbers below 1.

For instance, if k = 1/2, then the 1st to the 4th terms, inclusive, are greater than 1, and if k = 2, then the 6th to the 9th terms, inclusive, are greater than 1. Either way, there are 4 terms greater than 1. Statement two alone is sufficient to answer the question.

Answer: B