The step 3 is confusing how is there one in the denominator instead of i

Let's clarify Step 3.

The original Left-Hand Side (LHS) expression is:

$$\text{LHS} = \frac{1}{i + \frac{s'}{(1+s')^n - 1}}$$

In Step 2, we established that for this identity to hold, we assume $s' = i$.

Step 3 is about substituting $s'$ with $i$. This means we look for every instance of $s'$ in the expression and replace it with $i$.

Let's break down the LHS:

• The main numerator of the entire fraction is 1. There is no $s'$ in this part. So, it remains 1.
• The main denominator is $i + \frac{s'}{(1+s')^n - 1}$.
  • The first term in the main denominator is $i$. This is not $s'$, so it remains $i$.
  • The second term in the main denominator is a fraction: $\frac{s'}{(1+s')^n - 1}$.
    • The numerator of this smaller fraction is $s'$. This is where we substitute $s'$ with $i$. So, it becomes $i$.
    • The denominator of this smaller fraction is $(1+s')^n - 1$. Here, we substitute $s'$ with $i$. So, it becomes $(1+i)^n - 1$.

Putting it all together after the substitution:
The main numerator is 1.
The main denominator becomes $i + \frac{i}{(1+i)^n - 1}$.

So, the entire LHS after substitution in Step 3 is:

$$\text{LHS} = \frac{1}{i + \frac{i}{(1+i)^n - 1}}$$

The 1 in the main numerator is not an $s'$, so it does not change to $i$. The $i$ you are referring to is the $s'$ that was in the numerator of the nested fraction within the denominator.