if a,b,c are in A.P., prove that 8b^3-a^3-c^3=3ac(a+c)

c-b=b-a

2b=a+c

b=(a+c)/2

a^3+c^3-8b^3=a^3+c^3-8 ((a+c)/2)^3

=a^3+c^3 - 8((a^3+3a^2c+3ac^2+c^3)/8)  (since (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3)

=a^3+c^3-a^3-3a^2c-3ac^2-c^3

=-3a^2c-3ac^2

=-3ac(a+c)

Or we can say

8b³-a³- c³ = 3ac(a+c)