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Question
If a, b, c are in geometric progression, and if `"a"^(1/x) = "b"^(1/y) = "C"^(1/z)`, then prove that x, y, z are in arithmetic progression
Solution
Given a, b, c are in G.P.
⇒ b2 = ac
⇒ log b2 = log ac
(i.e.) 2log b = log a + log c .....(1)
We are given `"a"^(1/x) = "b"^(1/y) = "C"^(1/z)` = k ......(say)
⇒ log ak = `1/x`
log bk = `1/y`
log ck = `1/z`
⇒ log ak = `1/x`
⇒ x = log ka
y = log kb
z = log kc
Substituting these values in equation (1) we get
2y = x + z
⇒ x, y z are in A.P.
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