When the Sum of Num3ers Equals their Product [Part 3]

That is, a + b + … + n = a * b * … * n

When the Sum of Num3ers Equals their Product [Part 1]
2 + 2 = 2 * 2 = 4
etc.

When the Sum of Num3ers Equals their Product [Part 2]
3 + 3/2 = 3 * 3/2 = 9/2
3 + 3/2 + 9/7 = 3 * 3/2 + 9/7 = 3 * 3/2 * 9/7 = 81/14
3 + 3/2 + 9/7 + 81/67
= 3 * 3/2 + 9/7 + 81/67
= 3 * 3/2 * 9/7 + 81/67
= 3 * 3/2 * 9/7 * 81/67
= 6561/938
3 + 3/2 + 9/7 + 81/67 + 6561/5623
= 3 * 3/2 + 9/7 + 81/67 + 6561/5623
= 3 * 3/2 * 9/7 + 81/67 + 6561/5623
= 3 * 3/2 * 9/7 * 81/67 + 6561/5623
= 3 * 3/2 * 9/7 * 81/67 * 6561/5623
= 43046721/5274374

This is also true with 8 & 7:
8 + 8/7 = 8 * 8/7 = 64/7 = 9 + 1/7
8 * 8/7 * 64/57 = 8 + 8/7 + 64/57
8 * 8/7 * 64/57 = 4096/399
8 + 8/7 + 64/57 = 4096/399
8 * 8/7 * 64/57 * 4096/3697 = 16777216/1475103

It is also true wit 4 & 3:
4 + 4/3 = 4 * 4/3 = 16/3 = 5 + 1/3
etc.

We get similar results when combining    (5 & 4), (6 & 5), (7 & 6), …

n + n/(n-1) = n * n/(n-1) = n^2/(n-1), n > 1
n + n/(n-1) + x = n * n/(n-1) * x
n^2 – n + 1 ≠ 0,    x = n^2/(n^2 – n + 1),    n – 1 ≠ 0

If n = 3,    x = n^2/(n^2 – n + 1) = 9/7
If n = 8,    x = n^2/(n^2 – n + 1) = 64/57

[To Be Continued]