Leibniz+Harmonic+Triangle


 * __Leibniz Harmonic Triangle (For Interest Sake)__**



The Leibniz Harmonic Triangle starts off with each row beginning with the reciprocal of the row number. For example, row 4 would begin with one quarter, which is the reciprocal of the number 4. By looking at the starting fraction, one can determine the row number easily. Every fraction is calculated by looking at the sum of the two numbers just below it. The fractions in each row can be computed by subtracting. 1/5 – 1/6 = 1/30 etc. One characteristic of the Leibniz Harmonic Triangle is that it is symmetrical with respect to its vertical axis.

Just as Pascal's triangle can be computed using binomial coefficients, the Leibniz Harmonic Triangle can too with the below equation. In this equation, //c// refers to the column number and //r// refers to the row number.

__ The Leibniz Harmonic Triangle is shaped like the Pascal’s Triangle but there are some key differences: __  Instead of integers, the terms in the Leibniz Harmonic Triangle are fractions. Instead of adding the two terms above it to get the term below, a term is calculated by finding the difference of the term diagonally above it to the left and the term directly to the left of it. The seventh row has seven terms, unlike the Pascal's Triangle whose seventh row has eight terms. The reciprocal of the first term in each row in the Leibniz harmonic triangle will give the corresponding row number while the first term of the Pascal's Triangle will be number 1.

In each row, the starting term is the reciprocal of the row number. To obtain the terms for each row, just divide the initial term by the corresponding Pascal’s Triangle entries. The third row of the Leibniz Harmonic Triangle is to be divided by the second row from the top of the Pascal’s Triangle and so on.
 * __Obtaining the Leibniz Harmonic Triangle from Pascal’s Triangl e__ **

For Example:  The fraction ¼ is the initial term of the 4th row of the Leibniz Harmonic Triangle. Its corresponding row in the Pascal’s Triangle is the 3rd row with 1,3,3,1 as the terms of that row.

Divide ¼ by 1,3,3,1 respectively and obtain 4 terms. These 4 terms will form the 4th row of the Leibniz Harmonic Triangle which is ¼, 1/12, 1/12, ¼.

With that, this is the end of this learning package!