The circuit diagram shown consists of a large number of element (each element has two resistors $R_1$ and $R_2$). The resistance of the resistors in each subsequent element differs by $a$ factor of $K = 1/2$ from the resistance of the resistors in the previous elements. The equivalent reistance between $A$ and $B$ shown in figure is :
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When each element of circuit is multiplied by a factor $k$ then equivalent resistance also become $k$ times.
Let the equivalent resistance between $A$ and $B$ be $X$
So, equivalent circuit becomes
For $k=\frac{1}{2}$
$\Rightarrow \mathrm{X}=\frac{\left(\mathrm{R}_{1}-\mathrm{R}_{2}\right)+\sqrt{\mathrm{R}_{1}^{2}+\mathrm{R}_{2}^{2}+6 \mathrm{R}_{1} \mathrm{R}_{2}}}{2}$
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