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

Question

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

aceacharya · Answer

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

Bunuel · Answer

Similar question from OG13: Discussed here: an-arithmetic-sequence-is-a-sequence-in-which-each-term-59035.html

PareshGmat · Answer

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

shashankism · Answer

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.

whollyshiv · Answer

Let the numbers be -1, -3, -9, -27, -81 with r = 3
Check the option choices;
I. -2 , -6 , -18 , -54 ,-162 : GP with common ratio 3
II. 1 , -1 , -7 , -25 , -79 : Clearly not a G.P
III. Square root can’t be negative

Answer A.

Kab07 · Answer

for statement III where we are taking square roots 
what if the common ration is negative or the first term is negative of both the first term & common ratio are negative 
if we consider these cases the III collapse ?

Bunuel · Answer

Yes, it would have been better if the question had stated that the variables represent positive numbers. Then the expressions under the even roots would be defined.

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

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