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Q.
For every positive integer $n, 7^n - 3^n$ is divisible by
Principle of Mathematical Induction
Solution:
Let $P(n) : 7^n - 3^n$ is divisible by $4$.
For $n = 1,$
$P(1) : 7^1 - 3^1 = 4$, which is divisible by $4$. Thus, $P(n)$ is true for $n = 1.$
Let $P(k)$ be true for some natural number $k,$
i.e. $P(k) : 7^k - 3^k$ is divisible by $4.$
We can write $7^k - 3^k = 4d$, where $d \in N \,...\left(i\right)$
Now, we wish to prove that $P(k + 1)$ is true whenever P(k) is true, i.e. $7^{k + 1} - 3^{k + 1}$ is divisible by $4.$
Now, $7^{(k + 1)} - 3^{(k + 1)} = 7^{(k + 1)} - 7.3^k + 7.3^k - 3^{(k + 1)}$
$= 7(7^k - 3^k) + (7 - 3)3^k = 7(4d) + 4.3^k$ [using (i)]
$= 4(7d + 3^k)$, which is divisible by $4$.
Thus, $P(k + 1)$ is true whenever $P(k)$ is true. Therefore, by the principle of mathematical induction the statement is true for every positive integer $n$.