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Q.
If $10^{n} + 3\cdot4^{n+1} +k $ is divisible by $9$ for all $n \in N$, then the least positive integral value of $k$ is
Principle of Mathematical Induction
Solution:
Let $P\left(n\right) : 10^{n} + 3\cdot 4^{n+1} +k$ is divisible by $9\, \forall \,n \in N$.
For $n = 1$, we get
$ P \left(1 \right) = 10^{1} + 3 \cdot 4^{1+2}+ k $ is divisible by $9$
$ = 202 + k $ is divisible by $9$
So, the least value of $k$ is $5$.