Q. For all integers $n \geq 1,$ which of the following is divisible by 9?
A) $8^{n} + 1$
B) $4^{n} - 3n - 1$
C) $3^{2n} + 3n + 1$
D) $10^{n} + 1$
Explanation
(b)
∵ $4^{n} = (1 + 3)^{n}$
$= 1 + 3n + \frac{n(n - 1)}{2!}3^{2} + \ldots$
$\Longrightarrow \ 4^{n} - 3n - 1 = 3^{2}\left\lbrack \frac{n(n - 1)}{2!} + .. \right\rbrack$
It is clear from above that $4^{n} - 3n - 1$is divisible by 9.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.