First slide

Arithmetic progression

Question

If $\log_{y} x, \log_{z} y, - 15 \log_{x} z$ are in A.P., then

Moderate

A

$x = y^{- 3}$
B

$y = z^{- 2}$
C

$x = z^{3}$
D

None of these

Solution

Let d be the common difference of the A.P.
then, $\begin{array}{l} \log_{y} x = 1 + d \Rightarrow x = y^{1 + d} \\ \log_{z} y = 1 + 2 d \Rightarrow y = z^{1 + 2 d} \end{array}$
$\begin{array}{l} and - 15 \log_{x} z = 1 + 3 d \Rightarrow z = x^{- (1 + 3 d) / 15} \\ ∴ x = y^{1 + d} = z^{(1 + 2 d) (1 + d)} \\ = x^{- (1 + d) (1 + 2 d) (1 + 3 d) / 15} \\ \Rightarrow (1 + d) (1 + 2 d) (1 + 3 d) = - 15 \\ \Rightarrow 6 d^{3} + 11 d^{2} + 6 d + 16 = 0 \\ \Rightarrow (d + 2) (6 d^{2} - d + 8) = 0 \Rightarrow d = - 2 \\ ∴ x = y^{1 + d} = y^{- 1}, y = z^{1 + 2 d} = z^{- 3} and x = {(z^{- 3})}^{- 1} = z^{3} . \end{array}$

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