Consider a number n.
Out of all the fractions arising from the scenario by reducing some x from n and adding the ratio of the removed to the current n till n becomes 1, why does the sum be maximum if x = 1.
Lets say n = 5.
Why is 1/5+1/4+1/3+1/2 +1 = 2.28 always more than, alternative examples like. (Here x is always 1)
x = 2 , 1
2/5 + 1/3 + 1 = 1.73
or x = 3 , 1
3/5 + 1/2 + 1 = 2.1
or x = 4 , 1
4/5 + 1 = 1.8.
Inductively I can say this is true. How can I prove this ?
Notice that in every example, a sequence of fractions from the 'maximal' series is being replaced with a single fraction. In example 1, the sum $\frac{1}{5} + \frac{1}{4}$ is being replaced by the single fraction $\frac{2}{5}$. Similarly, in example 2, the sum $\frac{1}{5} + \frac{1}{4} + \frac{1}{3}$ is replaced by $\frac{3}{5}$, and in example 3, the sum $\frac{1}{5} + \frac{1}{4} + \frac{1}{3} + \frac{1}{2} $ is replaced by $\frac{4}{5}$.
But clearly in each case, the replacement lowers the sum, because
$$\frac{2}{5} = \frac{1}{5}+\frac{1}{5}<\frac{1}{5}+\frac{1}{4} \\ \frac{3}{5} = \frac{1}{5}+\frac{1}{5}+\frac{1}{5}<\frac{1}{5}+\frac{1}{4}+\frac{1}{3} \\ \frac{4}{5} = \frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}<\frac{1}{5}+\frac{1}{4}+\frac{1}{3}+\frac{1}{2}$$
This would be the case even if the reductions were out of order, e.g. 1,2,1 would lead to the series $\frac{1}{5}+\frac{2}{4}+\frac{1}{2}+1=2.2$; here the sum $\frac{1}{4}+\frac{1}{3}$ is being replaced by $\frac{2}{4}$, which will again lower the sum.