Problem Statement
Use the principle of mathematical induction to prove that $5^{2n} - 1$ is divisible by $24$ for all positive integers $n \in \mathbb{Z}^+$.
Verified Solution & Marking Scheme
Base Case: n = 1
For $$n = 1$$, $$5^{2(1)} - 1 = 5^2 - 1 = 25 - 1 = 24$$.
Since $$24 = 24 \times 1$$, the statement is true for $$n = 1$$.
Induction Hypothesis: Assume true for n = k
Assume that $$5^{2k} - 1 = 24m$$ for some integer $$m \in \mathbb{Z}$$.
This implies $$5^{2k} = 24m + 1$$.
Inductive Step: Prove for n = k + 1
Consider the expression for $$n = k + 1$$:
$$5^{2(k+1)} - 1 = 5^{2k + 2} - 1 = 5^2 \cdot 5^{2k} - 1 = 25 \cdot 5^{2k} - 1$$
Substitute $$5^{2k} = 24m + 1$$:
$$= 25(24m + 1) - 1 = 25(24m) + 25 - 1 = 25(24m) + 24 = 24(25m + 1)$$
Since $$m$$ is an integer, $$25m + 1$$ is an integer. Thus $$5^{2(k+1)} - 1$$ is divisible by $$24$$.
Formal Conclusion
Since the statement is true for $$n = 1$$, and whenever it is true for $$n = k$$ it is also true for $$n = k + 1$$, by the principle of mathematical induction the statement is true for all positive integers $$n \in \mathbb{Z}^+$$.