If ax = by = cz, x ≠ 0, y ≠ 0, z ≠ 0 and b2 = ac, then show that 
are in A.P.
Let ax = by = cz = k.
We know that if am = k, then a = 
.
⇒ a = 
, b = 
, c = 
Given, b2 = ac
⇒ 
= 
× 
We know that (am)n = amn and am × an = am + n.
⇒ 
= 
Bases are same, so we equate the powers.
⇒ 
= 
+ 
⇒ 
+ 
= 
+ 
⇒ 
- 
= 
- 
We know that when t1, t2, t3 … are in A.P., t3 – t2 = t2 – t1
∴
, 
and 
are in A.P.
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