${}^4\mathrm{C}_1+{}^5\mathrm{C}_2.\left(\frac{1}{2}\right)+{}^6\mathrm{C}_3.{\left(\frac{1}{2}\right)}^2+\dots \dots$ to $\infty$ terms:

$\begin{array}{rcl}\mathrm{S}& =& {}^4{\mathrm{C}}_1+{}^5{\mathrm{C}}_2 \left(\frac{1}{2}\right)+{}^6{\mathrm{C}}_3 {\left(\frac{1}{2}\right)}^2+....\infty \end{array}$

$\begin{array}{rcl}\frac{1}{2}\mathrm{S}& =& {}^4{\mathrm{C}}_1 \left(\frac{1}{2}\right)+{}^5{\mathrm{C}}_2 {\left(\frac{1}{2}\right)}^2+{}^6{\mathrm{C}}_3 {\left(\frac{1}{2}\right)}^3+....\infty \end{array}$

$\begin{array}{rcl}1+\frac{1}{2}\mathrm{S}& =& 1+{}^4{\mathrm{C}}_1 \left(\frac{1}{2}\right)+{}^5{\mathrm{C}}_2 {\left(\frac{1}{2}\right)}^2+....\infty \end{array}$

$\begin{array}{rcl}1+\frac{\mathrm{S}}{2}& =& {\left(1-\frac{1}{2}\right)}^{-4}=16\end{array}$

$\begin{array}{rcl}\mathrm{S}& =& 30\end{array}$