Problem of the Week

Problem 1 solution

This is a classic derivation of the golden ratio from its definition.

$\textbf{Method 1:}$
$\frac{a+b}{b} = \frac{b}{a}$
$\frac{a}{b} + \frac{b}{b} = \frac{b}{a}$
$\textbf{Let}\; x = \frac{b}{a}$
$\frac{1}{x} + 1 = x$
$1 + x = x^2$
$x^2 - x - 1 = 0$
$\textit{(Using Quadratic Formula)}\; x = \frac{1 \pm \sqrt{5}}{2}$
$\textbf{Note: } \frac{b}{a}>0, \quad \frac{b}{a} = \frac{1 + \sqrt{5}}{2} \approx 1.618$

$\textbf{Method 2:}$
$\frac{a+b}{b} = \frac{b}{a}$
$a^2 + ab = b^2$
$b^2 - ab -a^2 = 0$
$\left(b - \frac{a}{2}\right)^2 - \frac{5a^2}{2} = 0$
$b - \frac{a}{2} = \pm \frac{a\sqrt{5}}{2}$
$\textbf{Note: } b>0, \quad b = \frac{a+a\sqrt5}{2}$
$\frac{b}{a} = \frac{1+\sqrt{5}}{2} \approx 1.618$

Problem of the Week

Problem 1 solution

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