The next number in a certain sequence is defined by multiply

Question

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

Bunuel · Accepted Answer

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?

Given that:

\(a_2=a_1*k\)
\(a_3=a_1*k^2\)
\(a_4=a_1*k^3\)
...
\(a_n=a_1*k^{n-1}\)

Also given that k > 0, k ≠ 1 and n = 9.

(1) The ninth term in this sequence is 81:

\(a_9 = a_1 * k^8 = 81\). If \(a_1 = 1\), then all terms except \(a_1\) will be greater than 1. However, if \(a_1 = 2\), then all terms will be greater than 1. This information is not sufficient.

(2) The fifth term in this sequence is 1;

\(a_5 = a_1 * k^4 = 1\). Now, consider two cases:

• If \(a_1 < 1\), then \(k > 1\), and all terms from \(a_5\) onwards (\(a_6\), \(a_7\), \(a_8\), and \(a_9\)) will be greater than 1, resulting in 4 terms being greater than 1.
• If \(a_1 > 1\), then \(k < 1\), and all terms up to \(a_5\) (\(a_2\), \(a_3\), and \(a_4\)) will be greater than 1, again resulting in 4 terms being greater than 1.

This information is sufficient.

Answer: B.

EMPOWERgmatRichC · Answer

Hi All,

This is great 'concept' question. With a bit of 'playing around' you can probably find the patterns involved.

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 is 81

Since we don't know what the constant "K" is (it could be a positive fraction or a positive number > 1), we can TEST VALUES.

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 is 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 is 9 
Fact 1 is INSUFFICIENT.

Fact 2: The fifth term is 1.

Since the sequence either increases or decreases, we'd have...

4 numbers less than 1, then the number 1, then 4 numbers greater than 1

OR

4 numbers greater than 1, then the number 1, then 4 numbers less than 1

Regardless of which option, we end up with exactly 4 terms that are greater than 1.
Fact 2 is SUFFICIENT

Final Answer:  B

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

Bunuel · Answer

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AmritaSarkar89 · Answer

Just wondering,

can K take any positive fractional value or does it need to be an interger.

It does not explicitly mentions that K has to be an integer. I agree the answer will be the same in both the cases however just want to clarify in case we face a similar situation during the main exam.

EMPOWERgmatRichC · Answer

Hi AmritaSarkar89,

In my solution (above), I discuss this exact issue. GMAT questions are very carefully worded, so you have to pay attention to the details (what those details state and DON'T state). Here, we're told that K is a POSITIVE CONSTANT. That's does not mean that K is necessarily an integer (in the same way that stating N > 0 does not mean that N is necessarily an integer). Be mindful of these details when working through GMAT questions and make sure to write everything on the pad when you're working (so that you don't forget anything).

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

ldelcerro · Answer

But what if a1=-1 and K=2? or a1=0?

EMPOWERgmatRichC · Answer

Hi ldelcerro,

While it's certainly beneficial to think about all the possibilities (including negatives and 0), neither of those options 'fits' the given information in the two Facts. If the 9th term is 81, then the first term cannot be negative and it cannot be 0 (since repeatedly multiplying either of those options by a positive constant will NEVER lead to +81). The same issue occurs if the 5th term is 1. Thus, that initial term MUST be a positive.

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

BillyZ · Answer

Hi  Bunuel ,

Could you help to elaborate the highlighted part by giving an example to illustrate?

Bunuel · Answer

If  a_1=1 , then from  a_9=a_1*k^8=81  -->  1*k^8=81  -->  k=\sqrt{3}  -->  a_2=a_1*k=\sqrt{3}>1  and so on.

If  a_1=2 , then from  a_9=a_1*k^8=81  -->  2*k^8=81  -->  k \approx 1.6  -->  a_2=a_1*k \approx 3.3 > 1  and so on.

BillyZ · Answer

Hi  Bunuel , I think  a_5  should not count in the term as in highlighted part since  a_5=1 . If  a_1>1 ,  a_5=1 , then ( a_1 ,  a_2 ,  a_3 ,  a_4 ) will be > 1.

How could we calculate the value of k without using the calculator?  2*k^8=81  -->  k^8=40.5  -->   k \approx 1.6

Bunuel · Answer

We don't need exact value. From  2*k^8=81  it should be clear that k will be greater than 1, which is basically what we want to know. All further conclusions could be made based only on this.

roastedchips · Answer

Hi Bunuel, I don't quite understand the explanation for statement 2. You mention " if a1>1, then k<1, and all terms till a5(a2, a3 and a4), so again 4 terms will be greater than 1. Sufficient."
Why would K be <1? In the question it is clearly stated that K is a positive constant and does not equal 1. Thus, K will always be more than 1.

Please correct me if I am wrong. Thanks!

Bunuel · Answer

We are told that k is positive constant and k ≠ 1. So, 0 < k < 1 or k > 1. Nowhere we are told that k is an integer.

ccooley · Answer

I'm going to try to explain this one in a less 'mathematical' way, as an alternative for those of us who aren't super enthusiastic about algebra. Sequences are also pretty easy to translate into plain English, and that's how I prefer to think about them.

The question stem says that we're going to build a sequence by starting at some number, then multiplying by the same value over and over until we have 9 terms. For example, this would fit the rules:

1, 3, 9, 27, 81, 243, ... etc.

This would also fit the rules:

4, 2, 1, 1/2, 1/4, 1/8, 1/16, 1/32, ... etc.

This would  not  fit the rules, since k can't equal 1 (we can't just multiply by 1 over and over):

3, 3, 3, 3, 3, 3, ...

This also wouldn't work, since we have to  multiply  by the same value instead of adding the same value:

1, 2, 3, 4, 5, ...

So that's what the question is telling us. Notice that it's  not  telling us what the first number in the sequence is, so that could be literally anything. The first number could even be negative! (We just can't multiply by a negative number to get the next term.)

Finally, the question itself asks how many of the first 9 terms are greater than 1. I'm not sure how to figure that out mathematically, to be honest, so I'm going to focus on case testing.

Now, the statements.

Statement 1  The ninth term is 81.

_ _ _ _ _ _ _ _ 81

I know that I started with some value (which I don't know) and then multiplied by the same number over and over, until I hit 81 after multiplying 8 times.

In order to go  backwards , I'll have to start with the 81, and divide by the same number, over and over.

Suppose that k = 3. Build the sequence backwards:

_ _ _ _ _ _ _ 27 81

_ _ _ _ _ _ 9 27 81

_ _ _ _ _ 3 9 27 81

_ _ _ _ 1 3 9 27 81

The other four terms will be less than 1. So the answer to the question is "4 terms".

Now let's try a really crazy case. Suppose that k = 81. That is, we'll be dividing by 81 over and over.

_ _ _ _ _ _ _ 1 81

_ _ _ _ _ _ 1/81 1 81

1/81 is less than 1, and so are the other 6 terms at the beginning. The answer the question is "7 terms".

This statement is  not sufficient .

Statement 2 : The fifth term of the sequence is 1. So, it looks like this:

_ _ _ _ 1 _ _ _ _

We'll be going in both directions now. To go to the left, we'll divide by k. To go to the right, we'll multiply by k.

Let's say that k = 2.

_ _ 1/4 1/2 1 2 4 _ _

Interesting. The four to the left are less than 1. The answer to the question is "4".

Let's say that k = 1/2.

_ _ 4 2 1 1/2 1/4 _ _

Okay, now the four to the right are less than 1. The answer to the question is "4".

Will that always be true? Yeah - if k is a fraction, then the four numbers to the right will all be smaller than 1. If k isn't a fraction, the four numbers to the left will all be smaller than 1. That's because we're either dividing 1 by a number repeatedly (making it smaller), or multiplying 1 by a number repeatedly (making it bigger).

That's why this statement is sufficient,  even if k is a fraction.

ScottTargetTestPrep · Answer

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

zanaik89 · Answer

Hi Bunuel,

Why are we even considering k<1 as the question explicitly says pos constant...?

Also can u please give me an example where k>1 and a>1 gives a^5=1

Bunuel · Answer

1. We are told that k is positive constant and k ≠ 1. Nowhere we are told that k is an integer. So, 0 < k < 1 or k > 1. 2. I guess, you are talking about the second statement. a_5=a_1*k^4=1 cannot be true if both k and a1 are greater than 1. That's why the solution considers two cases: a_1<1 , k>1 and a_1>1 , k<1 .

Dani2325 · Answer

Hey, thanks for the explanation! I do get that but I am still wondering why a_0 isn’t considered here?

Bunuel · Answer

Don’t overthink it. If it started with a0, the first nine terms would be a0 to a8. The math doesn’t change.

The next number in a certain sequence is defined by multiply

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