Which term of the progression 18,-12,8, 512 729 ? - Vidyadip

Question

Which term of the progression $18,-12,8,\dots\text{is}\frac{512}{729}?$

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Answer

$18,-12,8,\dots\text{is}\frac{512}{729}$
$\text{a}=18,\text{n}=?,\text{t}_\text{n}=\frac{512}{729},\text{r}=\frac{\text{t}_{\text{n}-1}}{\text{t}_\text{n}}$
$\text{r}=\frac{\text{t}_2}{\text{t}_1}=\frac{-12}{18}=\frac{-2}{3}$
Also,
$\text{t}_\text{n}=\text{ar}^{\text{n}-1}$
$\frac{512}{729}=(18)\Big(\frac{-2}{3}\Big)^{\text{n}-1}$
$\frac{2^9}{36}\times\frac{1}{2\times3^2}=\Big(\frac{-2}{3}\Big)^{\text{n}-1}$
$\Big(\frac23\Big)^8=(-1)^{\text{n}-1}\Big(\frac23\Big)^{\text{n}-1}$
$\text{n}=9$

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