If-a-b-c-are-in-GP-and-log-c-a-log-b-c-log-a-b-are-in-AP-then-the-common-difference-of-the-AP-is-

Question Number 43602 by peter frank last updated on 12/Sep/18

If a, b, c are in GP and \log_{c} a, \log_{b} c, \log_{a} b

are in AP, then the common difference

of the AP is

Answered by $@ty@m last updated on 12/Sep/18

L e t \log_{c} a = x - y \Rightarrow a = c^{x - y} \dots (1)

\log_{b} c = x \Rightarrow b^{x} = c \dots (2)

\log_{a} b = x + y \Rightarrow a^{x + y} = b \dots (3)

F r o m (1) & (2),

a = b^{x (x - y)} \dots (4)

F r o m (3) & (4),

a = a^{(x + y) x (x - y)}

\Rightarrow (x + y) x (x - y) = 1 \dots (5)

∵ a, b & c a r e i n G . P .

∴ b^{2} = a c

\Rightarrow a^{2 (x + y)} = a . b^{x} f r o m (2) & (3),

\Rightarrow a^{2 (x + y)} = a . a^{x + y}

\Rightarrow a^{x + y} = a

\Rightarrow x + y = 1

\Rightarrow x = 1 - y

S u b s t i t u t i n g t h i s i n (5)

x (x - y) = 1

\Rightarrow (1 - y) (1 - 2 y) = 1

\Rightarrow 1 - 3 y + 2 y^{2} = 1

\Rightarrow y (2 y - 3) = 0

\Rightarrow y = 0 (i n a d m i s s i b l e)

o r y = \frac{3}{2} A n s .

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