Question

If $a,b,c$ are in A.P, then the value of the expression ${a^3} + {c^3} - 8{b^3}$ is equal to
$\left( a \right){\text{ 2abc}}$
$\left( b \right){\text{ 6abc}}$
$\left( c \right){\text{ 4abc}}$
$\left( d \right){\text{ - 6abc}}$

Answer 1

Hint:
Since, in the question, it is given that the given sequence, is in A.P so by using the formula of arithmetic mean which is given as $b = \dfrac{{a + c}}{2}$, and then we will substitute the value of $b$ in the expression ${a^3} + {c^3} - 8{b^3}$ and we will be able to find the value.
Formula used:
If $a,b,c$ are in arithmetic progression then,
Arithmetic mean,
$b = \dfrac{{a + c}}{2}$
Here,
$a,b,c$ , will be the terms of A.P

Complete step by step solution:
As we know that in arithmetic progression the arithmetic mean is found out by $b = \dfrac{{a + c}}{2}$
So in the given expression we will substitute the value of arithmetic mean which is $b$ , so we get
\[ \Rightarrow {a^3} + {c^3} - 8{\left( {\dfrac{{a + c}}{2}} \right)^3}\]
And as we know that ${\left( {x + y} \right)^3} = {x^3} + {y^3} + 3xy(x + y)$
So by using this formula in the above expression, we get
\[ \Rightarrow {a^3} + {c^3} - 8\left( {\dfrac{{{a^3} + {c^3} + 3ac(a + c)}}{8}} \right)\]
Since there is the same term present in the numerator and denominator so it will cancel out each other and we get
\[ \Rightarrow {a^3} + {c^3} - \left( {{a^3} + {c^3} + 3ac(a + c)} \right)\]
Now on solving furthermore, we get
\[ \Rightarrow {a^3} + c - {a^3} - {c^3} - 3ac(a + c)\]
Since here the same term or we can say the like term will cancel each other, so we get
\[ \Rightarrow - 3ac(2b)\]
And on solving the multiplication, we get
\[ \Rightarrow - 6abc\]
So the value of the expression ${a^3} + {c^3} - 8{b^3}$ , when $a,b,c$ are in A.P is $ - 6abc$ .

Hence, the option $\left( d \right)$ is correct.

Note:
This type of question is mainly based on the concept. So, whenever the arithmetic progression having the sequence of three numbers is given to us then we will use the formula of an arithmetic progression. And also while solving we have to be careful as the equations, might get mixed up while solving it. The difference between the sequences can also be represented by using $D$.