Parity Comparison in Binomial Expansions
Consider the binomial expansions of \((x+1)^n\) and \((x-1)^n\) where \(n\) is a positive integer. In part (a), provide the partial expansions up to the \(x^2\) term for both expressions. In part (b), based on your expansions, compare the behavior of the two expressions for even and odd values of \(n\).
Disproof by Counterexample: Sum of Two Odds
Prove or disprove: For every pair of odd integers, their sum is always an odd integer. Provide a counterexample if your claim is false.