$S=1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots$
$\frac{S}{3}=\frac{1}{3}+\frac{2}{3^{2}}+\frac{6}{3^{3}}+\ldots$
$\frac{2 S}{3}=1+\frac{1}{3}+\frac{4}{3^{2}}+\frac{4}{3^{3}}+\ldots$
$\frac{2 S}{3}=\frac{4}{3}+\frac{4}{3^{2}}+\frac{4}{3^{3}}+\ldots$
$S=\frac{3}{2}\left(\frac{4 / 3}{1-1 / 3}\right)=3$
Now, $\ell=(3)^{\log _{0.25}\left(\frac{1 / 3}{1-1 / 3}\right)}$
$\ell=3^{\log _{(1 / 4))}\left(\frac{1}{2}\right)}=3^{1 / 2}=\sqrt{3}$
$\Rightarrow \ell^{2}=3$
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