Use mathematical induction to prove that for all integers n

Use mathematical induction to prove that for all integers n greaterthanorequalto 0, 8n - 1 is divisible by 7.

Solution

Suppose the premise is true for some integer n>1, then there is an integer k such that
8 - 1 = 7k

8 = 7k + 1
8¹ = (8)8
= (7k+1)8
= 7(8k) + 8

Subtract 1 from both sides of the equation:
8¹ - 1 = 7(8k) + 8 - 1
8¹ - 1 = 7(8k) + 7
8¹ - 1 = 7(8k+1), which is a multiple of 7.

Therefore, if the premise is true for n, it is also true for n+1.