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Exercise 1.35

We need to verify that \(1 + 1/\varphi = \varphi\).

\(\varphi = \frac{1 + \sqrt{5}}{2}\) by definition (as described in Section 1.2.2).

Therefore we have, \[\begin{align} 1+1/\varphi &= 1 + \frac{1}{\frac{1+\sqrt{5}}{2}}\newline &= 1 + \frac{2}{1+\sqrt{5}}\newline&=1+\frac{2}{1+\sqrt{5}}\cdot \frac{1-\sqrt 5}{1-\sqrt 5}\newline &= 1 + \frac{\sqrt 5 - 1}{2}\newline&=\frac{1+\sqrt{5}}{2}\newline&=\varphi \end{align}\]