Prove that 5n - 1 is divisible by 4 using mathematical induction. | Discrete Structures 2075

Discrete Structures20755 marksmathematical Induction

Prove that $5^n - 1$ is divisible by 4 using mathematical induction. [5]

We want to prove that 4 (5ⁿ - 1) for all positive integers n, i.e., (5ⁿ - 1) is divisible by 4. --- Substitute n = 1: $5^1 - 1 = 5 - 1 = 4$ Since 4 is divisible by 4, the base case holds. --- Assume the statement is true for n = k, i.e., assume: $5^k - 1 \text{ is divisible by 4}$ This means ...

We want to prove that 4 (5ⁿ - 1) for all positive integers n, i.e., (5ⁿ - 1) is divisible by 4. --- Substitute n = 1: $$5^1 - 1 = 5 - 1 = 4$$ Since 4 is divisible by 4, the base case holds. --- Assume the statement is true for n = k, i.e., assume: $$5^k - 1 \text{ is divisible by 4}$$ This means ...

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