Question

a,b,c,d are 4 consecutive terms of a geometric sequence. Which of the following must be true?
I. b^2=ac
II. b/a=d/c
III. d/a=(c/b)^3

@Loujoelou @phi @Callisto

Answer 1

work with the definition of G.P.

re write a,b,c,d as
a, k a, k^2 a, k^3 a

and see what works

Answer 2

umm..

Answer 3

for choice A

replace b with k a
replace c with k^2 a
in
b^2 = ac

what do you get ?

Answer 4

ac=k^2 a^2
b=k^2 a^2
the same

Answer 5

so A is good.

try choice B

Answer 6

* choice II

Answer 7

b/a = ka/a = k
d/c = (k^3 a) / (k^2 /a) =k
the same

Answer 8

is there a typo? I see
II. b/d=d/c

but you did b/a

Answer 9

If II. is b/d = d/c it does not work.
if it is b/a = d/c it is true

did you try III ?

Answer 10

d/a = (k^3) /a
(c/b)^3 = (k^2a)/(ka) = k

they are different, aren't they?

Answer 11

you are getting sloppy

d = k^3 a
so
\[ \frac{d}{a} = \frac{k^3 a}{a} = \frac{k^3 \cancel{a}}{\cancel{a}} = k^3\]

what do you get for c/b ?

then raise that to the 3rd power because you want (c/b)^3

Answer 12

oh, yes, i miss it.
(c/b)^3 = k^3

Answer 13

so III. is true

Answer 14

all are true then

Answer 15

how about arithmetic sequence? the same way? (different formula)

Answer 16

are you saying II. has a typo, and is not correct as given in the question ?

Answer 17

why?
b/a = k
d/c = k...

Answer 18

yes, but the question asks II. b/d=d/c

Answer 19

ohh, i typed it wrongly.

Answer 20

ok

Answer 21

** how about arithmetic sequence? **
none of I. II. or III. will work for A.P.

Answer 22

nonono, i mean that's another question

Answer 23

If a,b,c,d are consecutive terms of an arithmetic sequence, which of the following must be true?
I. b-a=d-c
II. d,c,b,a are consecutive terms of an arithmetic sequence.
III. a<b<c<d

Answer 24

can you make this a new post. This one is too long.