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: 11k – 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 11k+1 – 6 is divisible by 5.
11k+1 – 6 = 11 * (11k) – 6
= 11* (6 + 5m) – 6 (now we use our hypothesis step, with rearranged expression 11k = 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.
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