For all integers $n \geq 1$, which of the following is divisible by 9 ?
For all integers $n \geq 1$, which of the following is divisible by 9 ?
$8^n+1$
$4^n-3 n-1$
$3^{2 n}+3 n+1$
$10^n+1$
Solution
$\because \quad 4^n=(1+3)^n$
$=1+3 n+\frac{n(n-1)}{2 !} 3^2+\ldots$
$\Rightarrow \quad 4^n-3 n-1=3^2\left[\frac{n(n-1)}{2 !}+\ldots\right]$
It is clear from above that $4^n-3 n-1$ is divisible by 9 .