A general Law of Mathematics

[BMAD]

Consider the table:

1 = 1

2+3+4 = 1 + 8

5+6+7+8+9 = 8 + 27

10+11+12+13+14+15+16 = 27 + 64

Guess the general law suggested by these examples. Prove it if you can.

[Barcallica]

Every row can be put as [x^2-(2x-2)]+[x^2-(2x-1)]+......+(x^2-2)+(x^2-1)+x^2=(x-1)^3+x^3 which converts to {[x^2-(2x-2)]+x^2}/2=(x-1)^3+x^3 thus can be proved after some easy simplification.

[Smash72]

Each sequence of numbers totals the last and next cube number.

17+18+19+20+21+22+23+24+25 = 64+125

26+27+28+29+30+31+32+33+34+35+36 = 125+216

[plasmid]

The general form of that pattern is

Sum from (x-1)²+1 to x² = (x-1)³ + x³

Make use of the fact that the sum of a series where each term increases or decreases by a fixed amount from the previous term can be rewritten as (the first term plus the last term) times (the number of terms in the series) divided by 2. If you didn't already know that, then the intuitive way of thinking about it is: (the first term plus the last term) is equal to (the second term plus the next-to-last term) which is equal to (the third term plus the third-from-the-end term) etc, so you can rearrange a sum of N numbers into N/2 pairs like that.

There are x²-(x-1)² terms on the left hand side (in general, if you count the whole numbers from n to m, then that's m-n+1 numbers), so we can change the left hand side to:

[First term plus last term] times [number of terms in the sequence] divided by 2

[(x-1)²+1 + x²] [x²-(x-1)²] / 2
Multiplying all that out gives
[(x²-2x+1) + 1 + x²] [x²-(x²-2x+1)] / 2
[2x²-2x+2] [2x-1] / 2
[x²-x+1] [2x-1]
2x³-3x²+3x-1

And that can be rewritten as
x³ + (x³-3x²+3x-1) = x³ + (x-1)³