Is tis always the case? Any algorithm to this/

1. Is it possible to express all unit fractions in this form?
2. Any algorithms that can be developed from such a relationship with unit fractions?

If you mean that one of the fractions should have a denominator that is one greater than the denominator that you start with, then a bit of algebra tells you that for fraction of 1/a you have:


\frac 1 a = \frac 1 {a+1} \ + \ ( \frac 1 a - \frac 1 {a+1} )= \frac 1 a + \frac 1 {a(a+1)}
\

Examples:

a = 3 gives 1/3 = 1/4 + 1/12
a = 4 gives 1/4 = 1/5 + 1/20
a = 5 gives 1/5 = 1/6+ 1/30