Method 1

Given that \( a, b, \) and \( c \) are in arithmetic progression (A.P), so we have the relationship:

\( b = \frac{a + c}{2} \)

\(\Rightarrow 2b=a+c\) ...(i)

We need to find the value of \( a^3 + c^3 - 8b^3 \).

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

Here \(a+c+(-2b)=2b-2b=0\)

\([ a+c=2b, \quad \text{Using (i)}]\)

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Therefore, \(a^3 + c^3 + (-2b)^3=3ac(-2b)=-6abc \).

[Here we used the identity \( x^3+y^3+z^3=3xyz\), if \(x+y+z=0\)]

Method 2

Given that \( a, b, \) and \( c \) are in arithmetic progression (A.P), so we have the relationship:

\( b = \frac{a + c}{2} \)

\(\Rightarrow 2b=a+c\) ...(i)

By taking cube on both sides of \( 2b = a + c \), we get:

\((2b)^3=(a+c)^3\)

\( 8b^3 = a^3 + c^3 + 3ac(a+c) \)

\( \Rightarrow a^3 + c^3 - 8b^3 = -3ac(a + c) \)

\( \Rightarrow a^3 + c^3 - 8b^3 = -3ac(2b)\).

\( [ a+c=2b,\quad \text{Using (i)}]\)

\( \Rightarrow a^3 + c^3 - 8b^3 = -6abc \)