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# A geometric sequence is a sequence in which each term after the first

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A geometric sequence is a sequence in which each term after the first  [#permalink]

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23 Apr 2013, 21:04
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Difficulty:

35% (medium)

Question Stats:

66% (01:17) correct 34% (01:18) wrong based on 140 sessions

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v, w, x, y, z
A geometric sequence is a sequence in which each term after the first is equal to the product of the preceding term and a constant. If the list of numbers shown above is an geometric sequence, which of the following must also be a geometric sequence?

I. 2v, 2w, 2x, 2y, 2z
II. v + 2, w + 2, x + 2, y + 2, z + 2
III. $$\sqrt{v}$$, $$\sqrt{w}$$, $$\sqrt{x}$$, $$\sqrt{y}$$, $$\sqrt{z}$$

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

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Re: A geometric sequence is a sequence in which each term after the first  [#permalink]

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23 Apr 2013, 22:21
1
Can be done quite easily if 4 nos are considered instead of the variables

For example

1, 2, 4, 8

I multiply 2 : 2, 4, 8, 16 ; they are still in GP
II Add 2: 3, 4, 6, 10 ; not in GP
III Square root : 1, \sqrt{2}, 2, 2\sqrt{2} ; still in GP

So clearly I and III i.e E is the correct answer
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Re: A geometric sequence is a sequence in which each term after the first  [#permalink]

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25 Apr 2013, 02:56
rakeshd347 wrote:
v, w, x, y, z

A geometric sequence is a sequence in which each term after the first is equal to the product of the preceding term and a constant. If the list of numbers shown above is an geometric sequence, which of the following must also be a geometric sequence?

I. 2v, 2w, 2x, 2y, 2z
II. v + 2, w + 2, x + 2, y + 2, z + 2
III. $$\sqrt{v}$$, $$\sqrt{w}$$, $$\sqrt{x}$$, $$\sqrt{y}$$, $$\sqrt{z}$$

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

Similar question from OG13:
Quote:
p, r, s, t, u

An arithmetic sequence is a sequence in which each term after the first term is equal to the sum of the preceding term and a constant. If the list of numbers shown above is an arithmetic sequence, which of the following must also be an arithmetic sequence?

I. 2p, 2r, 2s, 2t, 2u
II. p-3, r-3, s-3, t-3, u-3
III. p^2, r^2, s^2, t^2, u^2

(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III

Discussed here: an-arithmetic-sequence-is-a-sequence-in-which-each-term-59035.html
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Re: A geometric sequence is a sequence in which each term after the first  [#permalink]

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14 Jan 2015, 19:36
I. 2v, 2w, 2x, 2y, 2z

Multiplication by 2 keeps the GP sequence intact

II. v + 2, w + 2, x + 2, y + 2, z + 2

Addition breaks the GP sequence in this case

III.$$\sqrt{v}, \sqrt{w}, \sqrt{x},\sqrt{y}, \sqrt{z}$$

Square rooting is nothing but changing the power from 1 to $$\frac{1}{2}$$ ; it still keeps the GP sequence intact

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Re: A geometric sequence is a sequence in which each term after the first  [#permalink]

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05 Jul 2017, 05:17
mau5 wrote:
rakeshd347 wrote:
v, w, x, y, z
A geometric sequence is a sequence in which each term after
the first is equal to the product of the preceding term and a
constant. If the list of numbers shown above is an geometric
sequence, which of the following must also be a geometric
sequence?

I. 2v, 2w, 2x, 2y, 2z
II. v + 2, w + 2, x + 2, y + 2, z + 2
III.\sqrt{v}, \sqrt{w}, \sqrt{x},\sqrt{y}, \sqrt{z}

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

KUDOS please if you like my question.

We know that v,w,x,y,z are in GP. Thus, w/v = x/w = y/x = z/y = r(some constant, called the common ratio)

I. A multiplication by a constant (2) will not change the ratio, as evident.
III. The ratio for these terms will be another constant $$\sqrt{r}$$

E.

Let the common ratio of the terms in GP be r.
So, w = vr
x = vr^2
y = vr^3
z = vr^4

Now lets start checking I , II & III

I. 2v, 2w, 2x, 2y, 2z = 2v, 2vr, 2vr^2, 2vr^3, 2vr^4 Common ratio = r. So, GP
II. (v+2), (w+2), (x+2), (y+2), (z+2) = 2v+2, 2vr+2, 2vr^2 +2 , 2vr^3 +2, 2vr^4 +2 . Not in GP.
III. $$\sqrt{v},\sqrt{w},\sqrt{x},\sqrt{y},\sqrt{z}, = \sqrt{2v}, \sqrt{2vr}, \sqrt{2vr^2},\sqrt{2vr^3},\sqrt{2vr^4}$$

Common ration = $$\sqrt{r}$$. So, GP

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19 Mar 2020, 07:46
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Re: geometric sequence   [#permalink] 19 Mar 2020, 07:46