Answer
Solution and steps are provided below
Work Step by Step
let P(n) = 7^n − 1 is divisible by 6, for each integer n ≥ 0.
Basis step: Show that P(0) is true:
P(0) = 7^0 - 1 = 1-1 = 0, ( 7 | 0 is true (7 divides 0, so 0 is div by 7) )
Inductive step: Show that for all integers K ≥ 0, if P(K) is true then P(K+1) is true:
suppose P(K) = 7^k - 1 is div by 6, so 6 | 7^k, and so that 7^k = 6r for some integer r.
( That is if 7^k is div by 6, then 7^k is a multiple of 6).
[This is the inductive hypothesis]
P(K+1) = 7^(K+1) - 1 by definition
7^(k+1) - 1 = 7 ( 7^k ) -1
= 6 ( 7^k) + 7^k - 1
= 6 (7^k) + P(K)
= 6(7^k) + 6r
= 6 (7^k + r), and 7^k + r is an integer, so 6 (7^k +1) is div by 6
[And this is what we need to show]
so P(n) is true for n ≥ 0.
