Prove equality in formula related to logistic growth

Question

I'm trying to figure out the equality $$\frac{1}{y(1-y)}=\frac{1}{y-1}-\frac{1}{y}$$

I have tried but keep ending up with RHS $\frac{1}{y(y-1)}$.

Any help would be appreciated.

Answer 1

$$\frac{1}{y(y-1)}=\frac{y-(y-1)}{y(y-1)}=\frac{y}{y(y-1)}-\frac{y-1}{y(y-1)}=\frac{1}{y-1}-\frac{1}{y}$$

Answer 2

hint: write the top as $1 = y + (1-y)$ and split the fraction using the formula: $\dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}$ with $a = y, b = 1-y, c = y(1-y)$ , and simplify each fraction. Does this help ?

Answer 3

Set $y=2$ - your original formula gives $$\frac 1{-2}=1-\frac 12$$The left-hand side is negative and the right-hand side is positive, so the original statement in your question cannot be true. You seem to have proved that $$\frac 1{y(y-1)}=\frac 1{y-1}-\frac 1y$$ and this is indeed true where the fractions are defined.

All it is is a sign error, but substituting a simple value to check a formula is a useful thing to do where things seem to be going a bit wrong.