Creating the analogy with Pascal's arithmetic triangle, Leibniz formed harmonic triangle:
1/1
1/2 1/2
1/3 1/6 1/3
1/4 1/12 1/12 1/4
1/5 1/20 1/30 1/20 1/5
1/6 1/30 1/60 1/60 1/30 1/6
Right hand edge comprises of reciprocals of positive integers. (These are so called harmonic numbers.) Every number not on the edge is difference of two entries, 1 diagonally above and to right, and other immediately to right of number in question. Therefore, 1/20 = 1/12 - 1/30 = 1/4 - 1/5
i) Write next horizontal base of harmonic triangle.
ii) Prove that numbers in nth horizontal base of the triangle are reciprocals of numbers in corresponding base of arithmetic triangle divided by n+1, which is, if (n r] denotes rth entry in nth base of harmonic triangle,
(n r] = 1 / (n+1) (n r) , 0 less than or equal to r less than or equall to n
Utilize mathematical induction on n and recursion formula
( k + 1 r ] = ( k r ] - ( k+1 r+1]
iii) Illustrate that harmonic triangle is symmetric by verifying that ( n r] = ( n n-r] for 0 less than or equal to r less than or equal to n