Question 3: Prove that 5^n is divisible by 4 for each integer n ≥ 0 by mathematical induction.
Question 4: Prove that for all integers n ≥ 2, 2(3^n) - n(n+1).
Question 5: Prove that (1-4)^n - (1-4)^n-1 = -1 for all integers n ≥ 2.
Question 6: Prove that 2^n < (n + 2)^n for all integers n ≥ 0 by mathematical induction.
Question 7: A sequence f0, f1, f2 is defined as follows: f0 = 5, f1 = 16, and fn = 7fn-1 - 10fn-2 for all integers n ≥ 2. Prove that fn = 3(2^n) + 2(5^n) for all integers n ≥ 0.
Question 8: Compute 4^n for n = 1, 2, and make a conjecture about the units digit of 4^n where n is a positive integer. Prove the conjecture.