Magical properties of Cubes-2

We know that a sum of two cubes can be expressed as a product of two  numbers, where one of them is the sum of its root numbers. i.e., a³ + b³ = (a+b) (a² – ab + b² ) . When a = b, the product can be expressed as a palindromic series. It is illustrated with few examples.

 2³ + 2³ =16= 4 + 8 + 4 ( 3 terms)

3³ + 3³ =54= 6 + 12 + 18 + 12 + 6 (5 terms)

4³ + 4³ =128= 8 + 16 + 24 +32 + 24 + 16 + 8 ( 7 terms)

5³ + 5³ =250= 10 + 20 + 30 + 40 + 50 + 40 + 30 + 20 +10 (9 terms) and so on.

It is noted that the palidromic series begins with twice the root number (2a) in the given sum of two identical cubes a³ + a³ = 2a³ and contains 2a-1 terms.

When one of the root numbers is 1, then it shows a kind of arithmetic series. E.g.,

1³ + 2³ =9= 3 + 6 (=2x3)

1³+ 3³ = 28= 4 + 8 + 16 (=4x4)

1³ + 4³ = 65 = 5 + 10 + 20 + 30 (=6x5)

1³ + 5³ = 126 = 6 + 12 +24 + 36 + 48 (=8x6)

1³ + 6³ = 217 = 7 + 14 + 28 + 42 + 56 + 70  (=10x7) and so on.