5¹ − 1 = 4

Assume that 5^k − 1 is divisible by 4, therefore 5^k − 1 = 4A.

5^{k + 1} − 1 =
5(5)^k − 1 =
5(5)^k − 5 + 4 =
5(5^k − 1) + 4 =
20A + 4 = 4(5A + 1)
So 5^{k + 1} − 1 is a multiple of 4.

Hence, by mathematical induction we have proven that 5ⁿ − 1 is divisible by 4 for all natural numbers.