[tex]S_{15}=72+12+\cdots+\dfrac1{181398528}+\dfrac1{1088391168}[/tex]
[tex]S_{15}=72\left(\dfrac16\right)^0+72\left(\dfrac16\right)^1+\cdots+72\left(\dfrac16\right)^{13}+72\left(\dfrac16\right)^{14}[/tex]
[tex]S_{15}=72\displaystyle\sum_{i=1}^{15}\left(\frac16\right)^{i-1}[/tex]
[tex]\dfrac16S_{15}=72\displaystyle\sum_{i=1}^{15}\left(\frac16\right)^i[/tex]
[tex]\implies S_{15}-\dfrac16S_{15}=72\displaystyle\sum_{i=1}^{15}\bigg(\left(\dfrac16\right)^{i-1}-\left(\dfrac16\right)^i\bigg)[/tex]
[tex]\dfrac56S_{15}=72\bigg(\left(1-\dfrac16\right)+\left(\dfrac16-\dfrac1{6^2}\right)+\cdots+\left(\dfrac1{6^{13}}-\dfrac1{6^{14}}\right)+\left(\dfrac1{6^{14}}-\dfrac1{6^{15}}\right)\bigg)[/tex]
[tex]\dfrac56S_{15}=72\left(1-\dfrac1{6^{15}}\right)[/tex]
[tex]S_{15}=\dfrac{432}5\times\dfrac{6^{15}-1}{6^{15}}=86.4[/tex]
