Verify Lagrange's Formula

Question

If $a$ and $b$ are integers, (where $b$ is not $0$ or $1$), verify Lagrange's Formula:

$$a+\dfrac{1}{-b}=(a-1)+\dfrac{1}{1+\dfrac{1}{b-1}}$$

Answer 1

The key is to start from the right hand side

$ (a-1) + \frac{1}{1 + \frac{1}{b - 1}} $

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

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

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

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

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

$ = a - \frac{1}{b} $

$ = a + \frac{1}{-b} $

Answer 2

Move $a-1$ to the LHS to get

$$1-\frac1b=\frac1{1+\dfrac1{b-1}}.$$

Then

$$\frac{b-1}b=\frac1{\dfrac{b-1+1}{b-1}}.$$