If $a$ and $b$ are integers, (where $b$ is not $0$ or $1$), verify Lagrange's Formula:
$$a+\dfrac{1}{-b}=(a-1)+\dfrac{1}{1+\dfrac{1}{b-1}}$$
Verify Lagrange's Formula
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1Where is the difficulty? just combine the fractions on the right as indicated. – 2017-01-12
2 Answers
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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} $
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Move $a-1$ to the LHS to get
$$1-\frac1b=\frac1{1+\dfrac1{b-1}}.$$
Then
$$\frac{b-1}b=\frac1{\dfrac{b-1+1}{b-1}}.$$