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Is there a formula to add a sequence of cubes?


By examining the first five sums a remarkable discovery is suggested:

13 = 1
13+23=9
13+23+33=36
13+23+33+43=100
13+23+33+43+53=225

It seems that the sum is always square, but what is even more remarkable is that the sum of the first n cubes, 13+23+...+n3 = (n(n+1)/2)2, which is the square of the nth triangle number.

For example, 13+23+...+103=(10×11/2)2=552 = 3025.

Using a similar method used to prove that formula for the Sum of Squares, we shall prove this result deductively; it is hoped that it will offer some insight into how further the series of powers may be found.

Proof

n
r=1
r4–(r–1)4  =  n4–(n–1)4 + (n–1)4–(n–2)4 + ... + 34–24 + 24–14 + 14–04
     =  n4

But  r4–(r–1)4 = r4 – (r4–4r3+6r2–4r+1) = 4r3–6r2+4r–1.

4r3–6r2+4r–1  =  4 r3 – 6 r2 + 4 r1
   =  4 r3 – 6n(n+1)(2n+1)/6 + 4n(n+1)/2 – n
   =  4 r3n(n+1)(2n+1) + 2n(n+1) – n
   =  n4.
∴ 4 r3  =  n4 + n(n+1)(2n+1) – 2n(n+1) + n
   =  n(n3 + (n+1)(2n+1) – 2(n+1) + 1
   =  n(n3 + 2n2+3n+1 – 2n–2 + 1)
   =  n(n3+2n2+n)
   =  n2(n2+2n+1)
   =  n2(n+1)2
r3  =  n2(n+1)2/4
   =  (n(n+1)/2)2

In other words, the sum of the first n cubes is the square of the sum of the first n natural numbers.