Prove algebraically that (4n + 1)² − (2n − 1) is an even number for all positive integer values of n.
First of all expand the brackets and simplify the expression given:(4n+1)(4n+1)-(2n-1)= 8n2+8n+1-2n+1= 8n2+6n+2= 2(4n2+3n+1). Since the expression can be factorised with 2, and since 4n2+3n+1 will always be an integer since n is a positive integer, the expression is an even number for all positive integers n.
