5. 30pts] Prove or disprove each of the following claims: (a) For all positive integers, n is even if and only if 3n^2 + 8 is even (b) If a and b are rational numbers, then a^2 + b^2> 2ab (c) If n iss an even integer, then n + 1 is odd. (d) Every odd number is the difference of two perfect squares.

See explanation below

Step-by-step explanation:

(a) For all positive integers, n is even if and only if 3n^2 + 8 is even.

If n is even, then it can be written as n = 2k for some k,

⇔ 3n²+8 = 3 (2k) ²+8 ⇔3 (4k²) + 8 ⇔12k² + 8 ⇔ 2 (6k+4) which is even since it has the form 2r.

(b) If a and b are rational numbers, then a^2 + b^2> 2ab

We know that all squares are positive. Even the square of a negative number is positive ((-2) ² = 4).

Therefore, we can say that (a-b) ²≥0

⇒a²-2ab+b² ≥ 0

⇒a² + b² ≥ 2ab

(c) If n is an even integer, then n + 1 is odd.

If n is an even integer, then it can be written as n = 2k

Then, n + 1 would be (2k) + 1 which clearly is odd.

d) Every odd number is the difference of two perfect squares.

Let n = 2k + 1 for some k. Now take k² and (k+1) ².

The difference of these two numbers is

(k+1) ² - k² = k² + 2k + 1 - k² = 2k + 1, which is odd.

Therefore, every odd number is the difference of two perfect squares.