If \( a, b \) and \( c \) are in A.P, then \( a^3 + c^3 - 8b^3 \) is equal to:
Correct Answer: (d) \( -6abc \)
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 \(a+c+(-2b)=2b-2b=0\)
\([ a+c=2b, \quad \text{Using (i)}]\)
.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:
\((2b)^3=(a+c)^3\)
\( 8b^3 = a^3 + c^3 + 3ac(a+c) \)
\( \Rightarrow a^3 + c^3 - 8b^3 = -3ac(a + c) \)
\( \Rightarrow a^3 + c^3 - 8b^3 = -3ac(2b)\).
\( [ a+c=2b,\quad \text{Using (i)}]\)
\( \Rightarrow a^3 + c^3 - 8b^3 = -6abc \)