Question

Find S15 for the geometric series 72 + 12 + 2 + +…

Answer 1

$S_{15}=72+12+\cdots+\dfrac1{181398528}+\dfrac1{1088391168}$
$S_{15}=72\left(\dfrac16\right)^0+72\left(\dfrac16\right)^1+\cdots+72\left(\dfrac16\right)^{13}+72\left(\dfrac16\right)^{14}$
$S_{15}=72\displaystyle\sum_{i=1}^{15}\left(\frac16\right)^{i-1}$

$\dfrac16S_{15}=72\displaystyle\sum_{i=1}^{15}\left(\frac16\right)^i$

$\implies S_{15}-\dfrac16S_{15}=72\displaystyle\sum_{i=1}^{15}\bigg(\left(\dfrac16\right)^{i-1}-\left(\dfrac16\right)^i\bigg)$
$\dfrac56S_{15}=72\bigg(\left(1-\dfrac16\right)+\left(\dfrac16-\dfrac1{6^2}\right)+\cdots+\left(\dfrac1{6^{13}}-\dfrac1{6^{14}}\right)+\left(\dfrac1{6^{14}}-\dfrac1{6^{15}}\right)\bigg)$
$\dfrac56S_{15}=72\left(1-\dfrac1{6^{15}}\right)$
$S_{15}=\dfrac{432}5\times\dfrac{6^{15}-1}{6^{15}}=86.4$