extraordinary properties of products of any four successive numbers

The extraordinary properties of products of four successive numbers

The product of any four successive numbers with any given difference d  has some peculiar properties

d = 1

1x 2 x 3 x 4 = 24 + 1 = 25 = 5²

 2 x 3 x 4 x 5 = 120 + 1 = 121 = 11²

 3 x 4 x 5 x 6 = 360 + 1 = 361 = 19²

  4 x 5 x 6 x 7 = 840 + 1 = 841 = 29²

  5 x6 x7 x8 = 1680 + 1 = 1681 = 41²

d=2

1 x 3 x 5 x7 = 105 + 16 = 121 = 11²

2 x 4 x 6 x8 = 384 + 16 =  400 = 20²

3 x 5 x7 x9 = 945 + 16 = 961 = 31²

4 x 6 x 8 x10 = 1920 + 16 = 1936 = 44²

5 x7 x9 x11 = 3465 + 16 = 3481 = 59²

d = 3

1 x 4 x7 x 10 = 280 + 81 = 19²

2 x 5 x 8 x11 = 880 + 81 = 31²

3 x 6 x9 x12 = 1944 + 81 = 45²

4 x7 x 10 x 13 = 3640 + 81 = 61²

5 x 8 x 11 x 14 = 6160 + 81 = 79²

The square root number is found to be the mean of  products of extreme numbers  a (n+3d) and middle numbers  (n +d ) (n + 2d) and is equal to (n² + 3nd + d²)² The number added with the product of four successive numbers is equal to d^4..  It gives the following algebraic relation.

n (n+d) (n+2d) (n+3d) + d⁴  =  (n² + 3nd + d²)²