The equivalent resistance between points $A$ and $B$ of an infinite network of resistances each of $1\,\Omega $ connected as shown, is
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(c) By formula $R = {R_1} + \frac{{{R_2} \times R}}{{{R_2} + R}}$
$R = 1 + \frac{{1 \times R}}{{1 + R}}$
$==>$ ${R^2} + R = 1 + R + R$
$==>$ ${R^2} - R - 1 = 0$ or $R = \frac{{1 \pm \sqrt {1 + 4} }}{2}$ $ = \frac{{1 \pm \sqrt 5 }}{2}$
Since $R$ cannot be negative, hence $R = \frac{{1 + \sqrt 5 }}{2}\,\Omega $
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