Equivalent resistance between the adjacent corners of a regular $n$-sided polygon of uniform wire of resistance $R$ would be:
JEE MAIN 2023, Medium
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Suppose resistance of each arm is $r$, then $r = R / n$
$R _{ eq(AB )}=\frac{ R _1 R_2}{R_1+R_2}$
$\frac{ r ( n -1) r }{ r +( n -1) r }$
$=\frac{ r ( n -1) r }{ nr }$
$=\frac{ n -1}{ n } r$
$=\frac{( n -1) R }{ n ^2}$
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