Prove that the square of an odd number is always 1 more than a multiple of 4

2n+1 will always be an odd number (e.g. if n is equal to 3 the answer would be 7, an odd number) So, we square 2n+1 and write this as (2n+1)² 2n +12n 4n² 2n+1 2n 1Then multiple out the brackets to give 4n²+4n+1 We then put the equation into brackets again 4(n² + n) +1 The 4(n² + n) term will aways be a multiple of 4Therefore we have proved that:(2n+1)² = 4(n² + n) +1 and therefore have proved that the square of an odd number is always 1 more than a multiple of 4.