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Otrada [13]
3 years ago
11

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

Mathematics
1 answer:
Sedbober [7]3 years ago
6 0
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
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