Let P(n) be the statement that 11^n - 6 is divisible by 5.
BASE CASE:
Let n = 1. This gives 11^1 - 6 = 5, obviously divisible by 5, therefore we know that P(1) is true.
HYPOTHESIS STEP:
Assume that P(k) is true for some positive integer k.
We can write this a different way: 11^k – 6 = 5m where m is also a positive integer.
INDUCTION STEP:
We will now show that P(k+1) is true.
P(k+1) states that 11^k+1 – 6 is divisible by 5.
11^k+1 – 6 = 11 * (11^k) – 6
= 11* (6 + 5m) – 6 (now we use our hypothesis step, with rearranged expression 11^k = 6 + 5m)
= 55m + 60 (multiplying out the brackets gives)
= 5 * (11m + 12) (now factorising again)
Which shows that this is a factor of 5 and that P(k+1) is true.
CONCLUSION: Since P(k+1) is true given P(k), and we know that P(1) is true, we have proved by induction that P(n) is true for all positive integer n.