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

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mau5
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A geometric sequence is a sequence in which each term after the first [#permalink] New post   23 Apr 2013, 22:04
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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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E
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Re: A geometric sequence is a sequence in which each term after the first [#permalink] New post   23 Apr 2013, 23:21
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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] New post   25 Apr 2013, 03:56
Expert Reply
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] New post   14 Jan 2015, 20:36
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

Answer = E
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Re: A geometric sequence is a sequence in which each term after the first [#permalink] New post   05 Jul 2017, 06:17
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

Answer E.
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Re: A geometric sequence is a sequence in which each term after the first [#permalink] New post   18 Mar 2024, 12:00
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