Prove algebraically that the square of any odd number is always also an odd number.

Firstly, an algebraic expression of an odd number should be identified, such as 2n+1 or 2n-1. Doing this would also indicate the knowledge that 2n is always an even number, which will be important further on. This should then be written out as (2n+1)(2n+1). Multiplying these two expressions together gives us 4n² +2n + 2n +1, or 4n² +4n +1.In order to prove this is odd, we can simply take a factor of 2 out of the first 2 terms to leave us with 2(2n² +2n) +1. If we now refer to 2n² +2n as x, we can rewrite this equation as 2x +1, which is the same algebraic expression we used to identify a number as odd. We can thus deduce that the square of any odd number is also always odd.