Question

For all integers $n\ge 1$, which of the following is divisible by $9$?

• A. $8^n+1$
• B. $4^n-3n-1$
• C. $3^{2n}+3n+1$
• D. ${10}^n+1$

Answer 1

The correct option is B

$4^n-3n-1$

Finding the divisibility:

Since we can write

$\begin{array}{rcl}{4}^n& =& {(3+1)}^n\\ & =& {}^{n}C_0{3}^n+{}^{n}C_1{3}^{n-1}+{}^{n}C_2{3}^{n-2}+.....+{}^{n}C_{n-1}3+{}^{n}C_n[expandingu\sin gbinamialexpansion]\\ & =& 3^n+n{3}^{n-1}+.....+3n+1\\ 4^n-3n-1& =& (3^n+n{3}^{n-1}+.....)\\ 4^n-3n-1& =& 3^2(3^{n-2}+n{3}^{n-3}+.....)\\ 4^n-3n-1& =& 9(3^{n-2}+n{3}^{n-3}+.....)\end{array}$

Therefore $4^n-3n-1$is multiple of $9$

Hence, correct option is $\mathbf{\left(}\mathit{B}\mathbf{\right)}$