If-a-b-and-c-are-in-a-GP-Prove-that-log-n-a-log-n-b-log-n-c-are-in-AP-

Question Number 42670 by Tawa1 last updated on 31/Aug/18

If a, b and c are in a GP . Prove that \log_{n} a, \log_{n} b, \log_{n} c are in AP

Commented by maxmathsup by imad last updated on 31/Aug/18

w e h a v e \frac{b}{a} = \frac{c}{b} \Rightarrow b^{2} = a c \Rightarrow 2 l n (b) = l n (a) + l n (c) \Rightarrow

\frac{2 l n (b)}{l n (n)} = \frac{l n (a)}{l n (n)} + \frac{l n (c)}{l n (n)} \Rightarrow 2 {l o g}_{n} (b) = {l o g}_{n} (a) + {l o g}_{n} (c) \Rightarrow

{l o g}_{n} (b) - {l o g}_{n} (a) = {l o g}_{n} (c) - l o g (b) (= r) \Rightarrow {l o g}_{n} (a), {l o g}_{n} (b) {a n d l o g}_{n} (c) a r e i n

A P .

Commented by Tawa1 last updated on 31/Aug/18

God bless you sir

Commented by prof Abdo imad last updated on 31/Aug/18

t h a n k y o u s i r .

Answered by Rio Michael last updated on 31/Aug/18

a, b, c i n G P

b^{2} = a c

t a k i n g {l o g}_{n} o n b o t h s i d e s

{l o g}_{n} b^{2} = {l o g}_{n} a c

2 {l o g}_{n} b = {l o g}_{n} a + {l o g}_{n} c

{l o g}_{n} b = \frac{{l o g}_{n} a + {l o g}_{n} c}{2}

w h i c h i s t h e A r i t h m e t i c m e a n (A M) [b = \frac{a + c}{2}] f o r A . P s

Commented by Tawa1 last updated on 31/Aug/18

God bless you sir

Commented by Rio Michael last updated on 31/Aug/18

w e l c o m e