Make the correct alternative in the following question:
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).

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