Question

if a,b,c are in GP,prove that 1/a,1/b,1/c are also in GP

Answer 1

if a,b,c are in GP, so b/a = c/b or b/c = a/b
now, if 1/a,1/b,1/c are also in GP, so must be satisfies 1/b * a = 1/c * b or a/b = b/c.
proof

Answer 2

Gp = geometric progression

Answer 3

Generally speaking if a^n = b^(n+k) then the inverse of each side reveals
1/a^n = 1/b^(n+k)
Or
(1/a)^n = (1/b)^(n+k)

If they are right next to each other in progression then k = 1.

Answer 4

i am not understand @RadEn

Answer 5

i am not understand

Answer 6

each ratio of geo series is always same right ?

Answer 7

yes and

Answer 8

if given a,b,c, so the ratio (r) = b/a = c/b right ?

Answer 9

right and b=ar & c=ar^2
but then

Answer 10

i dont think b=ar and c=ar^2
but, from b/a = c/b, it means also b/c = a/b ?

Answer 11

\[\frac{ b }{ a }=\frac{ c }{ b }=r\]

Answer 12

b/a = c/b ---> b * b = a * c ----> b/c = a/b, agree ?

Answer 13

yes

Answer 14

now, do same idea for series of 1/a, 1/b, 1/c
r = (1/b)/(1/a) = (1/c)/(1/b)
simplify and u must show to get b/c = a/b too

Answer 15

what! ummm...I'm not even there when it comes to proofs. I'm just a beginner.

Answer 16

thanks to all.................