1. Give a proof by contradiction that if 3n + 2 is odd, then n is odd.
2. Prove that √3 is an irrational number.
3. Show that for all positive n ∈ ℕ, the following formula is true:
1^2 + 2^2 + 3^2 + ... + n^2 = n(n + 1)(2n + 1) / 6.
4. Find and prove a formula for the following sum in terms of n:
1 + 5 + 9 + ... + (4n - 3).
5. Suppose that a0, a1, a2, ... is a sequence defined as follows:
a0 = 12, a1 = 29
ak = 5ak-1 - 6ak-2
for all integers k ≥ 2. Use induction to prove that
an = 5 ⋅ 3^n + 7 ⋅ 2^n
for all integers n ≥ 0.
6. If a > 1 is a positive integer, show that there exist two different natural numbers k and ℓ for which a^k - a^ℓ is divisible by 10.
[Hint: use the Pigeonhole principle. What should the pigeonholes be here?]