Question

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

Answer 1

`S_(15)=72+12+\cdots+\frac1{181398528}+\frac1{1088391168}`

`S_(15)=72\left(\frac16\right)^0+72\left(\frac16\right)^1+\cdots+72\left(\frac16\right)^(13)+72\left(\frac16\right)^(14)`

`S_(15)=72\displaystyle\sum_(i=1)^(15)\left(\frac16\right)^(i-1)`


`\frac16S_(15)=72\displaystyle\sum_(i=1)^(15)\left(\frac16\right)^i`


`\implies S_(15)-\frac16S_(15)=72\displaystyle\sum_(i=1)^(15)\bigg(\left(\frac16\right)^(i-1)-\left(\frac16\right)^i\bigg)`

`\frac56S_(15)=72\bigg(\left(1-\frac16\right)+\left(\frac16-\frac1{6^2}\right)+\cdots+\left(\frac1{6^(13)}-\frac1{6^(14)}\right)+\left(\frac1{6^(14)}-\frac1{6^(15)}\right)\bigg)`

`\frac56S_(15)=72\left(1-\frac1{6^(15)}\right)`

`S_(15)=\frac{432}5*(6^(15)-1)/(6^(15))=86.4`