(a) $47^n+16 n-1$
$\begin{array}{r}
(48-1)^n+4(4 n)-1 \\
48 k+(-1)^n+4 m-1 \\
4(12 k+m)+(-1)^n-1
\end{array}$
$47^n+16 n-1$ is divisible by 4 if $n$ is odd.
$\therefore$ It is false for all values of $n$
(b) $2\left(4^{2 n+1}\right)-3^{3 n+1}$
$8 \cdot 4^{2 n}-3 \cdot 3^{14}$
Put $n=1, \quad 8 \cdot 4^2-3 \cdot 3^1=8 \times 16-9=119$
119 is not divisible by 9 .
(c) $4^{n+}-3 n-1$
Put $n=2, \quad 4^2-3(2)-1=16-7=9$ which is not divisible by 11 .
(d) $3 \cdot 5^{2 n+1}+2^{3 n+1}$
$\begin{gathered}
15 \cdot 5^{2 n}+2 \cdot 2^{3 n} \\
15 \cdot\left(25^n+2 \cdot(8)^n\right. \\
15 \cdot(17+8)^n+2 \cdot 8^n \\
15 \cdot\left(17 k+8^n\right)+2 \cdot 8^n \\
15\left(17 k+8^n\right)+8^n(15+2)
\end{gathered}$
$=17\left(15 k+8^n\right)$
$\therefore 3 \cdot 5^{3 n+1}+2^{3 n+1}$ is divisible by 17 for all values of $n$