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HARMONIC TRIANGLE

Harmonic Triangle is a triangle wherein the first line is the harmonic series, and each fraction in the triangle is the sum of the term to the right of it, and below it.

  1 1/2 1/3 1/4 1/5 1/6 ...
  1/2 1/6 1/12 1/20 1/30 ...  
  1/3 1/12 1/30 1/60 ...    
  1/4 1/20 1/60 ...      
  1/5 1/30 ...        
  1/6 ...          
               
               

It can be noted that each term on the harmonic triangle is a fraction.  Furthermore, each fraction is a reciprocal of a natural number. 

To build the harmonic triangle, simply fill the first row and column with the harmonic series.  Then fill in the values of the remaining cells.  Each of this cell has a value which is equal to the number above it minus the number diagonally above it on the right.  For example, the value on the 2nd row and 2nd column is 1/6.  This number is equal to 1/2-1/3, number above it (1/2) minus the number diagonally above it on the right (1/3).

A special property of the harmonic triangle is that each fractions on the harmonic triangle is equal to the sum of all the fraction below it and its right.  For example, the second term on the first row, 1/2, is equal to the sum of the series beginning with the number below it.

                              1/2 = 1/6 + 1/12 + 1/20 + 1/30  .....

 

In Summary, the harmonic triangle has the following properties:

1) The first line, as we know, is the harmonic series.

2) The fractions on the second line are one half the reciprocls of triangular numbers.  The fractions on the second line sums up to one.

3) The fractions on the third line are 1/3 the reciprocals of the pyramidal numbers.

4) Each fractions on the harmonic triangle is equal to the sum of the fraction below it, and diagonally below it on the left.

5) Each fractions on the harmonic triangle is equal to the difference of the fraction aboveit, and diagonally above it on the right.

6) Each fractions on the harmonic triangle is equal to the sum of all the fraction below it and its right. 

 

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