A geometric sequence is one in which the ratio of any term after the first to the preceding term is a constant. If the letters a, b, c, d represent a geometric sequence in normal alphabetical order, which of the following must also represent a geometric sequence for all values of k?
I. dk, ck, bk, ak
II. a+k, b+2k, c+3k, d+4k
III. ak^4, bk^3, ck^2, dk
1. I only
2.I and II only
3.II and III only
4.I and III only
5.I, II, and III
Can somebody please help me with this problem?
the answer is I and III only
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The G.P. described (a,b,c,d) can be written as
a , ar , ar^2 , ar^3
where r is the constant ratio which is multiplied.
1. dk, ck, bk, ak
becomes ak , akr , akr^2 , akr^3
which is still a G.P. Hence True
2. a+k, b+2k, c+3k, d+4k
The above form is an Arithmetic Progression and cannot be expressed in terms of a & r. Hence Untrue.
3. ak^4, bk^3, ck^2, dk
can be rephrased to dk , ck^2, bk^3, ak^4
which is in G.P. form. Hence true.
Answer - 4
a , ar , ar^2 , ar^3
where r is the constant ratio which is multiplied.
1. dk, ck, bk, ak
becomes ak , akr , akr^2 , akr^3
which is still a G.P. Hence True
2. a+k, b+2k, c+3k, d+4k
The above form is an Arithmetic Progression and cannot be expressed in terms of a & r. Hence Untrue.
3. ak^4, bk^3, ck^2, dk
can be rephrased to dk , ck^2, bk^3, ak^4
which is in G.P. form. Hence true.
Answer - 4
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Great question! This one is interesting because it defines the term "geometric sequence" in the question, so you don't have to have prior knowledge of this sequence type to answer the question. However, if you do have prior knowledge, it will make figuring out the question much easier.
It reminds me of PS 100 in OG 13 - if you know what a logarithmic scale is, you're set. If not, you can still figure the question out, but it may take additional time to comprehend.
Back to this question. Do you remember what a geometric sequence is?
Here's an example: 3, 9, 27, 81...
Let's see if this example works with the definition given in the question: "the ratio of any term after the first to the preceding term is a constant" in a geometric sequence. If we pick the third term, 27, and find the ratio of this to the preceding term, 9, we get 3. If repeat this process for the next term, 81, with the preceding term, 27, we also get 3. So we see that this ratio is constant for the sequence above.
As we expect from the GMAT, this definition is not the simplest way to state what is happening in a geometric sequence. This is where some prior knowledge would come in handy. What are we really saying if the ratio between subsequent terms is constant?
We're saying that to get the next term in the sequence, we multiply the current term by the constant. To get to the third term in the sequence above, we multiply the second term, 9, by the constant, 3, giving us 27. That's how a sequence earns the status of "geometric". Much easier to think about, right?
So, if a, b, c, and d represent a geometric sequence, then a times some constant is b, b times the same constant is c, and c times the same constant is d.
Now, looking at statements I-III, we've got some things to think about to answer the question. What sorts of changes will maintain the sequence's status as geometric? What sorts of changes will change the sequence's status?
I see three major changes in this group:
1. Reversing the order of terms (Statement 1)
2. Multiplying by a constant or powers of a constant (Statements 1 and 3)
3. Adding a constant or multiples of a constant to the terms (Statement 2)
Let's investigate each of these changes and see how it affects the sequence's status using our example sequence: 3, 9, 27, 81...
#1: If we reverse the order of terms, we get 81, 27, 9, 3... Is this still a geometric sequence?
Instead of multiplying by 3 to get each subsequent term as we did above, now we're dividing by 3. But dividing by 3 is the same as multiplying by 1/3, so we've still got a geometric sequence. This change preserves the sequence's status as geometric.
#2: If we multiply all of the terms by a constant, let's say 2, then our sequence becomes 6, 18, 54, 162... Is this still a geometric sequence?
Well, notice that each subsequent term is still a factor of 3 away from the previous term, so multiplying by a constant does not change the geometric status of the sequence.
What about if we multiply by a power of a constant, like in Statement 3? Let's try it: 3 * 2^4, 9 * 2^3, 27 * 2^2, 81 * 2 yields 48, 72, 108, 162. Is this still a geometric sequence?
It's not readily evident, but let's check: 72/48 = 9/6 = 3/2, and 108/72 = 12/8 = 3/2. Yes, it's still a geometric sequence - we're multiplying each term by 3/2 to get the next.
Here's why: Going from 3 to 9 in the first two terms means we're multiplying by 3, but going from 2^4 to 2^3 means we're dividing by 2. If we think of it as all multiplication, we're really multiplying by 3/2.
So, if we're multiplying by a constant or power of a constant, we're preserving the geometric status of the sequence. That tells us Statements 1 and 3 are true for the question and eliminates the first three answer choices.
Take a second to let this sink in. A geometric sequence is multiplicative, created by multiplying the same constant with each term to get the next new term. If we adjust the sequence by multiplying by a different constant, it preserves the geometric nature of the sequence. The only time it wouldn't is if we were multiplying each term by a different constant.
Now for #3: If we add a constant, let's say 2, to all of the terms, the our sequence becomes 5, 11, 29, 83. Is this still a geometric sequence?
Well, it can't be - all of these numbers are prime! In other words, in order to get from 5 to 11, we'd multiply by 11/5; in order to get from 11 to 29, we'd multiply by 29/11. We're no longer multiplying by a constant between terms here, so we've broken the geometric status of the sequence.
The same holds true if we add multiples of a constant to the sequence. Using 2 as our constant again, let's try what we're given in Statement 2: 3 + 2, 9 + 2*2, 27 + 3*2, 81 + 4*2 gives us 5, 13, 33, 89. To get from 5 to 13, we'd multiply by 13/5; to get from 13 to 33, we'd multiply by 33/13. No dice.
Answer choice 5 is out, and we end up with 4.
So, what's the takeaway from the problem? Two things.
First, be familiar with arithmetic and geometric sequence. Know how their rules work and what sorts of changes you can make to a sequence that preserve its status as arithmetic or geometric and what sorts of changes remove its status.
Second, feel free to test numbers for a question like this. It often won't prove your answer definitively, but it does allow you to explore what's going on in the question and abstract general rules from there.
Hope this helps!
It reminds me of PS 100 in OG 13 - if you know what a logarithmic scale is, you're set. If not, you can still figure the question out, but it may take additional time to comprehend.
Back to this question. Do you remember what a geometric sequence is?
Here's an example: 3, 9, 27, 81...
Let's see if this example works with the definition given in the question: "the ratio of any term after the first to the preceding term is a constant" in a geometric sequence. If we pick the third term, 27, and find the ratio of this to the preceding term, 9, we get 3. If repeat this process for the next term, 81, with the preceding term, 27, we also get 3. So we see that this ratio is constant for the sequence above.
As we expect from the GMAT, this definition is not the simplest way to state what is happening in a geometric sequence. This is where some prior knowledge would come in handy. What are we really saying if the ratio between subsequent terms is constant?
We're saying that to get the next term in the sequence, we multiply the current term by the constant. To get to the third term in the sequence above, we multiply the second term, 9, by the constant, 3, giving us 27. That's how a sequence earns the status of "geometric". Much easier to think about, right?
So, if a, b, c, and d represent a geometric sequence, then a times some constant is b, b times the same constant is c, and c times the same constant is d.
Now, looking at statements I-III, we've got some things to think about to answer the question. What sorts of changes will maintain the sequence's status as geometric? What sorts of changes will change the sequence's status?
I see three major changes in this group:
1. Reversing the order of terms (Statement 1)
2. Multiplying by a constant or powers of a constant (Statements 1 and 3)
3. Adding a constant or multiples of a constant to the terms (Statement 2)
Let's investigate each of these changes and see how it affects the sequence's status using our example sequence: 3, 9, 27, 81...
#1: If we reverse the order of terms, we get 81, 27, 9, 3... Is this still a geometric sequence?
Instead of multiplying by 3 to get each subsequent term as we did above, now we're dividing by 3. But dividing by 3 is the same as multiplying by 1/3, so we've still got a geometric sequence. This change preserves the sequence's status as geometric.
#2: If we multiply all of the terms by a constant, let's say 2, then our sequence becomes 6, 18, 54, 162... Is this still a geometric sequence?
Well, notice that each subsequent term is still a factor of 3 away from the previous term, so multiplying by a constant does not change the geometric status of the sequence.
What about if we multiply by a power of a constant, like in Statement 3? Let's try it: 3 * 2^4, 9 * 2^3, 27 * 2^2, 81 * 2 yields 48, 72, 108, 162. Is this still a geometric sequence?
It's not readily evident, but let's check: 72/48 = 9/6 = 3/2, and 108/72 = 12/8 = 3/2. Yes, it's still a geometric sequence - we're multiplying each term by 3/2 to get the next.
Here's why: Going from 3 to 9 in the first two terms means we're multiplying by 3, but going from 2^4 to 2^3 means we're dividing by 2. If we think of it as all multiplication, we're really multiplying by 3/2.
So, if we're multiplying by a constant or power of a constant, we're preserving the geometric status of the sequence. That tells us Statements 1 and 3 are true for the question and eliminates the first three answer choices.
Take a second to let this sink in. A geometric sequence is multiplicative, created by multiplying the same constant with each term to get the next new term. If we adjust the sequence by multiplying by a different constant, it preserves the geometric nature of the sequence. The only time it wouldn't is if we were multiplying each term by a different constant.
Now for #3: If we add a constant, let's say 2, to all of the terms, the our sequence becomes 5, 11, 29, 83. Is this still a geometric sequence?
Well, it can't be - all of these numbers are prime! In other words, in order to get from 5 to 11, we'd multiply by 11/5; in order to get from 11 to 29, we'd multiply by 29/11. We're no longer multiplying by a constant between terms here, so we've broken the geometric status of the sequence.
The same holds true if we add multiples of a constant to the sequence. Using 2 as our constant again, let's try what we're given in Statement 2: 3 + 2, 9 + 2*2, 27 + 3*2, 81 + 4*2 gives us 5, 13, 33, 89. To get from 5 to 13, we'd multiply by 13/5; to get from 13 to 33, we'd multiply by 33/13. No dice.
Answer choice 5 is out, and we end up with 4.
So, what's the takeaway from the problem? Two things.
First, be familiar with arithmetic and geometric sequence. Know how their rules work and what sorts of changes you can make to a sequence that preserve its status as arithmetic or geometric and what sorts of changes remove its status.
Second, feel free to test numbers for a question like this. It often won't prove your answer definitively, but it does allow you to explore what's going on in the question and abstract general rules from there.
Hope this helps!
