Prove by induction that 11^n - 6 is divisible by 5 for all positive integer n.

Let P(n) be the statement that 11ⁿ - 6 is divisible by 5.

BASE CASE: Let n = 1.  This gives 11¹ - 6 = 5, obviously divisible by 5, therefore we know that P(1) is true.

HYPOTHOSIS 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.