Question

If 1, log_y x, log_z y, –15 log_x z are in A.P., then-

• A. z³ = x
• B. x = y^–2
• C. z^–2 = y
• D. None of these

Answer 1

Answer: (A)

z³ = x

Answer 2

Let d be the common difference. Then

log_y x = 1 + d ̃ x = y^1+d

log_z y = 1 + 2d ̃ y = z^1+2d

and –15log_x z = 1 + 3d ̃ z = x^–(1+3d)/15

x = y^1+d = z^{(1 + 2d) (1 + d)}

= x^{–(1 + d)(1 + 2d) (1 + 3d)/15}

̃ (1 + d) (1 + 2d) (1 + 3d) = –15

̃ 6d³ + 11d² + 6d + 16 = 0

̃ (d + 2) (6d² – d + 8) = 0 ̃ d = –2

\ x = y^1+d = y^–1, y = z^{1– 4} = z^–3

and x = (z^–3)^–1 = z³.

Hence (1) is the correct answer.