Q. If $a_{k}$ is the coefficient of${\ x}^{k}$ in the expansion of ${(1 + x + x^{2})}^{n}$ for $k = 0,\ 1,2,\$…., $2n$ then
A) $- a_{0}$
B) $3^{n}$
C) $n \bullet 3^{n + 1}$
D) $n \bullet 3^{n}$
Explanation
(d)
We have,
$\left( 1 + x + x^{2} \right)^{n} = a_{0} + a_{1}x + a_{2}x^{3} +$….$+ a_{2n}x^{2n}$
On differentiating both sides, we get
$n(1 + x + x^{2})^{n - 1}(1 + 2x) = a_{1} + 2a_{2}x + 3a_{3}x^{2}$
+…$+ 2na_{2n}x^{2n - 1}$
Now, on putting $x = 1$, we get
$n({3)}^{n - 1} \bullet (3) = a_{1} + 2a_{2} + 3a_{3} +$...$+ 2na_{2n}$
$\Longrightarrow a_{1} + 2a_{2} + 3a_{3} +$….$+ 2na_{2n} = n \bullet 3^{n}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.