Go to Fun_Math Content Table Sums of Integers and Series

Sum of cube - 1

Sum of cube - 2

Sum of odd integers,starting with 1, gives the square as shown in the diagram below.

1 = 1One way to interpret this pattern is^{2}1 + 3 = 2^{2}1 + 3 + 5 = 3^{2}1 + 3 + 5 + 7 = 4^{2}1 + 3 + 5 + 7 + 9 = 5^{2}1 + 3 + 5 + 7 + 9 + 11 = 6^{2}

Therefore the sum of the row,which is the sum of odd integers, is NxN=N

This is what we has been discussed in the previous section on integer.

Fibonacci(1170-1250) arranged the same odd numbers in a different pattern as shown below

and came up with a very elegant proof regarding the sum of integers cubed.

1 = 1Fibonacci's interpretation of this table is the same as above.^{3}3 + 5 = 2^{3}7 + 9 + 11 = 3^{3}13 + 15 + 17 + 19 = 4^{3}21 + 23 + 25 + 27 + 29 = 5^{3}31 + 33 + 35 + 37 + 39 + 41 = 6^{3}

So the N-th row has N terms and their average is N

Therefore the sum of the row is N

Now let us take a look at the left hand side.

The last term of the odd integer on the N-th row is a triangular number T_{N},

because T_{N} = 1 + 2 + .. + N

And we know that sum of the first p odd integers is p^{2}

Therefore Fibonacci concluded

1^{3}+ 2^{3}+ 3^{3}+ ... + N^{3} = (1 + 2 + 3 + ... + N)^{2} =
(T_{N})^{2} = {(1/2)N(N+1)}^{2}

Compare the figure below with Fibonacci's diagram. The average numbers in Fibonacci's diagram

are now represented by squares in the diagonal position.

You can see the process in animation for N=6 case.

It is easy to figure out the following relation.

1^{3} + 2^{3} + 3^{3} + 4^{3} + .. + N^{3} = (1 + 2 + 3 + 4 + .. + N)^{2}

Please recall the "Triangular Number"

T_{n} = 1 + 2 + 3 + ... + n = (1/2)N(N+1)

So Sum of cube from 1 to N = (1/4){N(N+1)}^{2}

**To create this drawing and animation: **
** Load sumcube_2.lsp ---> (load "sumcube_2")
Then from command line, type sumcube_2 **

**************

Now let us separate upper left portion of this tile pattern as shown in the left below.

If this is rotated 90 degrees clock-wise, and aligned with the horizontal line,

where numbers 1,2,3,4 & 5 are written ,then we have a staircase like pattern

as shown in the right.

Just as the original square rotated 90 degrees around the red tile 4 times forms a square.

so does the staircase . This will lead to the next visualization scheme.

***************sumcube_2_1.dwg** ************ **************sumcube_2_2.dwg** **************

which is then copied around 4 times, will form a square.

From the figure it is easy to see

4 x Sum = {N (N + 1)}

Therefore Sum = (1/4){N (N + 1)}

You can see the process in animation

for N=4 case.

**To create this drawing and animation: **
** Load sumcube_1.lsp ---> (load "sumcube_1")
Then from command line, type sumcube_1 **

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- Wells, David: The Penguin Dictionary of Curious and Interesting GEOMETRY. London,England: Penguin Books, p.198, 1991.(Out of print)
- Conway,J.H., Guy,R.K.: The Book of Numbers. Springer-Verlag,New York, p 27, 1995.
- Nelson,R.B. : Proofs Without Words: Exercises in Visual Thinking. MAA,p.85,87, 1993.
- Dantzig, Tobias : NUMBER, The Language of Science. New York: The Free Press, p.269, 1930.

Go to Fun_Math Content Table Sums of Integers and Series

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Last Updated July 9-th, 2006

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