The resultant resistance of the infinite circuit made of resistors, each of resistance r, between A and B is nearly

By symmetry equivalent circuit

 
 
 $$\begin{gathered} \frac{(R_{eq}+2r)\times 2r}{R_{eq}+4r}=R_{eq} \\ 2r{R}_{eq}+4r^2=R_{eq}^2+4r{R}_{eq} \\ {R}_{eq}^2+2r{R}_{eq}-4r^2=0R_{eq}=\frac{-2r\pm \sqrt{4r^2+16r^2}}{2} \\ =\frac{-2r\pm \sqrt{20r^2}}{2}=\frac{-2r\pm 4.5r}{2}=1.25r \end{gathered}$$