Adding Sequence of a/b+b/c+c/d+...+n+1/n

firstly some rules:

a/b+b/a = (a²+b²)/ab
a/b+b/c+c/a = (a²c+ab²+bc²)/abc
a/b+b/c+c/d+d/a = (a²cd+ab²d+abc²+bcd²)/abcd
a/b+b/c+c/d+d/e+e/a = (a²cde+ab²de+abc²e+abcd²+bcde²)/abcde

those procedurs are no so difficult, then I'll try to write the next considering an algorithm:

a/b+b/c+c/d+d/e+e/f+f/a = (a²cdef+ab²def+abc²ef+abcd²f+abcde²+bcdef²)/abcdef

as you can see there are some tips to done it. At first, lets conceptualize them:

A¹+B²+C³+...+N^k = (A^s+1+B^s+2+C^s+3+...+N^s+k)/k

1) the number of variables matches the number of constituents in the result numerator (1, 2, 3, ..., k = s+1, s+2, s+3, ..., s+k)
2) in the result denominator, write all the variables (multiplying)
3) in the result numerator, fill each constituent with all the variables
4) then, square the top variable from 1 in the s+1 constituent
5) erase the down variable from 1 in the s+1 constituent