Question

If $a,b,c$ are in A.P., then $a^2+4b^3+c^3$ is equal to ______.

• A. $3b(a^2+c^2)$
• B. $3b(a^2-c^2)$
• C. $3b(c^2-a^2)$
• D. $3b^2(a^2+c^2)$

Answer 1

The correct option is A $3b(a^2+c^2)$

If $a,b,c$ are in A.P., then

$a^3+4b^3+c^3=a^3+4{\left(\frac{a+c}{2}\right)}^3+c^3$

$=a^3+c^3+\frac{1}{2}(a+c{)}^3$

$=\frac{3}{2}(a^3+c^3+a^{2}c+a{c}^2)$

$=3\left(\frac{a+c}{2}\right)(a^2+c^2)$

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