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User:Dylanwhs/dylwhs 2

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The numbers which are subtracted from the Root form a series

If one looks at the sum, the number of terms relates to the cube root of the sum itself.

1:~ 1 2:~ 1 + 7 3:~ 1 + 7 + 19 4:~ 1 + 7 + 19 + 37 5:~ 1 + 7 + 19 + 37 + 61 . . n:~ 1 + 7 + 19 + 37 + 61 + ....... +

Notice also that these terms can be written as

n:~ 1 + (1+6) + (1+18) + (1+36) + (1+60) + .... +

  = 1 + (1 + 6) + (1 +6 + 12) + (1+6+12+18) + (1+6+12+18+24) + ....
  = 1 + (1 + A) + (1 + B) + (1 + C) + (1 + D) +

Where

A = 6 B = 6 + 12 C = 6 + 12 + 18 D = 6 + 12 + 18 + 24

1 occurs n times
6 = 6.1 occurs n-1 times

12 = 6.2 occurs n-2 times 18 = 6.3 occurs n-3 times 24 = 6.4 occurs n-4 times

The term which occurs 1 times only will be 6.(n-1)

Thus the sum becomes

Therefore: