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# A geometric sequence is one in which the ratio of any term after the f

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Math Expert
Joined: 02 Sep 2009
Posts: 51067
A geometric sequence is one in which the ratio of any term after the f  [#permalink]

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11 Nov 2014, 08:16
1
9
00:00

Difficulty:

45% (medium)

Question Stats:

62% (01:47) correct 38% (01:52) wrong based on 178 sessions

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Tough and Tricky questions: Sequences.

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. $$ak^4$$, $$bk^3$$, $$ck^2$$, $$dk$$

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II, and III

Kudos for a correct solution.

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Re: A geometric sequence is one in which the ratio of any term after the f  [#permalink]

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11 Nov 2014, 11:37
1
Bunuel wrote:

Tough and Tricky questions: Sequences.

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. $$ak^4$$, $$bk^3$$, $$ck^2$$, $$dk$$

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II, and III

Kudos for a correct solution.

Note that if a,b,c and d are in GP then... b^2=ac and c^2=bd

Option 1: $$dk$$, $$ck$$, $$bk$$, $$ak$$ so we have c^2k^2=dk*bk...Sufficient 1 is GP...Option C ruled out
Option 3 is better bet so let's do that before: b^2*k^6=ak^4*ck^2 or b^2*k^6=ack^6...St 2 is also GP.... Option A and B are out

Option 2: $$a + k$$, $$b + 2k$$, $$c + 3k$$, $$d + 4k$$

(b+2k)^2=(a+k)*(c+k) or b^2+4k^2+4k=ac+ak+ck+k^2 ---->3k^2 =(a+c-4)k... or k(3k-(a+c-4))=0 Clearly it does not meet criteria for GP..

Ans is D
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Re: A geometric sequence is one in which the ratio of any term after the f  [#permalink]

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Updated on: 11 Nov 2014, 15:34
1
1
Somehow I feel that i got tricked
First, a geometric sequence has to be on this form ax ,bx, cx... ( x is the common factor(ratio))
so 1) is definitely correct
3) dividing the sequence will make it easy to understand ak^4, bk^3 is a geometric sequence with K as a common factor. The same would be for the rest ck^2, dk^1
So 3) is correct
2) is a straight arithmetic sequence
hence D is the answer (unless i got tricked since the title is tricky questions )
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Originally posted by clipea12 on 11 Nov 2014, 14:53.
Last edited by clipea12 on 11 Nov 2014, 15:34, edited 1 time in total.
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Re: A geometric sequence is one in which the ratio of any term after the f  [#permalink]

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11 Nov 2014, 23:41
2
a, b, c, d are in GA, then

$$\frac{b}{a} = \frac{c}{b} = \frac{d}{c} = Constant$$

$$b^2 = ac$$

$$c^2 = bd$$

Sequence II fails. I & III are sure winner

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Re: A geometric sequence is one in which the ratio of any term after the f  [#permalink]

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12 Nov 2014, 02:58
Official Solution:

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. $$ak^4$$, $$bk^3$$, $$ck^2$$, $$dk$$

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II, and III

Our first task is to ensure that we understand the definition of a geometric sequence. Let's use the sequence given to us: $$a$$, $$b$$, $$c$$, $$d$$. We are told that the ratio of any term after the first to the preceding term is a constant. In other words, $$\frac{b}{a} = \text{some constant}$$, which is the same constant for the other ratios ($$\frac{c}{b}$$ and $$\frac{d}{c}$$). Let's name that constant $$r$$. Thus, we have the following:
$$\frac{b}{a} = \frac{c}{b} = \frac{d}{c} = r$$

By a series of substitutions, we can rewrite the sequence in terms of just $$a$$ and $$r$$:

$$a$$, $$b$$, $$c$$, $$d$$ is the same as $$a$$, $$ar$$, $$ar^2$$, $$ar^3$$

Rewriting the sequence this way highlights the role of the constant ratio $$r$$. That is, to move forward in the sequence one step, we just multiply by a constant factor $$r$$. Rewriting also lets us substitute into the alternative sequences and watch what happens.

I. $$dk$$, $$ck$$, $$bk$$, $$ak$$

This sequence is the same as $$ar^3k$$, $$ar^2k$$, $$ark$$, $$ak$$. To move forward in the sequence, we divide by $$r$$. This is the same thing as multiplying by $$\frac{1}{r}$$. Since this factor is constant throughout the sequence, the sequence is geometric.

II. $$a + k$$, $$b + 2k$$, $$c + 3k$$, $$d + 4k$$

This sequence is the same as $$a + k$$, $$ar + 2k$$, $$ar^2 + 3k$$, $$ar^3 + 4k$$. To move forward in this sequence, we cannot simply multiply by a constant expression. The presence of the plus sign means that we will not have a constant ratio between successive terms, and this sequence is not geometric.

III. $$ak^4$$, $$bk^3$$, $$ck^2$$, $$dk$$

This sequence is the same as $$ak^4$$, $$ark^3$$, $$ar^2k^2$$, $$ar^3k$$. To move forward in the sequence, we multiply by $$r$$ and divide by $$k$$. In other words, we multiply by $$\frac{r}{k}$$, which is a constant factor. This sequence is therefore geometric.

Only sequences I and III are geometric.

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Re: A geometric sequence is one in which the ratio of any term after the f  [#permalink]

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14 Aug 2018, 11:35
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Re: A geometric sequence is one in which the ratio of any term after the f &nbs [#permalink] 14 Aug 2018, 11:35
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