Question:
Make the correct alternative in the following question:
If $10^{n}+3 \times 4^{n+2}+\lambda$ is divisible by 9 for all $n \in \mathbf{N}$, then the least positive integral value of $\lambda$ is
(a) 5
(b) 3
(c) 7
(d) 1
Solution:
Let $\mathrm{P}(n): 10^{n}+3 \times 4^{n+2}+\lambda$ be divisible by 9 for all $n \in \mathbf{N}$.
For $n=1$,
$\mathrm{P}(1)=10^{1}+3 \times 4^{1+2}+\lambda$
$=10+3 \times 4^{3}+\lambda$
$=10+192+\lambda$
$=202+\lambda$
As, the least value of $P(1)$ which is divisible by 9 is 207 .
$\Rightarrow 202+\lambda=207$
$\Rightarrow \lambda=207-202$
$\therefore \lambda=5$
Hence, the correct alternative is option (a).