Divisibility proofs by induction

For each of the statements below, the corresponding activity will lead you to explore a proof by induction of the statement.

• Divisibility proof 1. For the sequence a_n = a_n-1 + 2n with a₁ = 1, a_n is always odd.
• Divisibility proof 2. For the sequence a_n = 5a_n-1 + 4 with a₁ = 3, a_n is always of the form 4K + 3.
• Divisibility proof 3. For the sequence a_n = 2a_n-1 + a_n-2 with a₁ = 1 and a₂ = 2, the sum n + a_n is always even.
• Divisibility proof 4. For value of n² + n is always even.