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  4. If n ∈ N, then 72n + 23n -3. 3 n - 1 is always divisible by

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Q. If $n \in N$, then $7^{2n} + 2^{3n -3}. 3^{ n - 1}$ is always divisible by

Principle of Mathematical Induction

Solution:

Putting $n = 1$ in $ 7^{2n} + 2^{3n - 3} .3^{n - 1}$
then, $7^{2 \times 1} + 2^{3\times 1 -3}. 3^{1- 1}$
$= 7^2 + 2^0.3^0 = 49 + 1 = 50 ...(i)$
Also, $n = 2$
$7^{2 \times 2} + 2^{3\times 2- 3} .3^{2 - 1} = 2401 + 24 = 2425 ...(ii)$
From (i) and (ii) it is always divisible by $25$.