CPM Homework Help : PC Problem 8-151

Use mathematical induction to prove that 5ⁿ − 1 is divisible by 4. Homework Help ✎

Step 1:

5¹ − 1 = 4

Step 2:

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

Step 3:

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.

Step 4:

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