If ln (a + c) , ln (c – a), ln (a – 2b + c) are in A.P., then :

The correct answer is: a, b, c are in H.P.

a,b,c are in HP

Ln(a-2b+c)-ln(c-a)=ln(c-a)-ln(a+c)
Ln((a-2b+c)/(c-a))=ln((c-a)/(a+c))
(a-2b+c)/(c-a)=(c-a)/(a+c)
(a-2b+c)(a+c)=(c-a)(c-a)
a²-+ac-2ab-2bc+ac+c²=c²+a²-2ac
on solving, we get
2ac = ab+bc
On dividing the above equation by abc, we get
2/b = 1/c+ 1/a or,
1/c- 1/b = 1/b- 1/a
This is the property of an HP.
Hence, a,b ,c are in  HP.

an A.P is a sequence of mathematical terms which have a common difference with their adjacent elements
An = A1+(n-1)x d
A2= A1 + d
A3 = A1 + 2d
A3 = A2+d
A3-A2=A2-A1