Question

Prove for all natural numbers n if n is odd then n^2 + 3n is even

Answer 1

an odd number can be written in the form 2x+1.

(2x+1)(2x+1)+3(2x+1) =
4x²+4x+1+6x+3=
4x² + 10x +4

now 4x² is even because 4 is a multiple of 2
10x is even because 10 is a multiple of 2
4 is even

add 3 evens together and you get yet another even