A galvanometer of resistance ' G ', is shunted by a resistance ' S...

Let the original resistance of the galvanometer be $G$.

When the galvanometer is shunted by a resistance $S$, the equivalent resistance of the parallel combination is $\dfrac{GS}{G+S}$.

Let $R$ be the resistance connected in series with this combination.

To keep the main current in the circuit unchanged, the total resistance of the modified galvanometer must be equal to its original resistance $G$.

$R + \dfrac{GS}{G+S} = G$

$R = G - \dfrac{GS}{G+S}$

$R = \dfrac{G(G+S) - GS}{G+S}$

$R = \dfrac{G^2 + GS - GS}{G+S}$

$R = \dfrac{G^2}{S+G}$

Answer: $\dfrac{G^2}{S+G}$