Question

If the geometric sequences $162,54,18,....$ and $\frac{2}{81},\frac{2}{27},\frac{2}{9},....$ have their $n^{th}$ term equal, find the value of $n$

Answer 1

The first geometric progression is $162,54,18,......$ where the first term is $a_1=164$, second term is $a_2=54$ and so on.

We find the common ratio $r$ by dividing the second term by first term as shown below:

$r=\frac{54}{162}=\frac{1}{3}$

We know that the general term of an geometric progression with first term $a$ and common ratio $r$ is $T_n=a{r}^{n-1}$, therefore, the $n$th term of the first G.P is:

$T_n=a{r}^{n-1}=162{\left(\frac{1}{3}\right)}^{n-1}=162{\left(3^{-1}\right)}^{n-1}=162\times {\left(3\right)}^{-n+1}.........\left(1\right)$

Similarly, the second geometric progression is $\frac{2}{81},\frac{2}{27},\frac{2}{9},.........$ where the first term is $a_1=\frac{2}{81}$, second term is $a_2=\frac{2}{27}$ and so on.

We find the common ratio $r$ by dividing the second term by first term as shown below:

$r=\frac{\frac{2}{27}}{\frac{2}{81}}=\frac{2}{27}\times \frac{81}{2}=3$

We know that the general term of an geometric progression with first term $a$ and common ratio $r$ is $T_n=a{r}^{n-1}$, therefore, the $n$th term of the second A.P is:

$T_n=a{r}^{n-1}=\frac{2}{81}\times {\left(3\right)}^{n-1}.........\left(2\right)$

Now, since it is given that the $n$th terms of the two G.P's are equal therefore, equating equations 1 and 2 we get

$162\times {\left(3\right)}^{-n+1}=\frac{2}{81}\times {\left(3\right)}^{n-1}\Rightarrow 162\times \frac{81}{2}{\left(3\right)}^{-n+1}={\left(3\right)}^{n-1}\Rightarrow (81\times 81){\left(3\right)}^{-n+1}={\left(3\right)}^{n-1}\Rightarrow (3^4\times 3^4){\left(3\right)}^{-n+1}={\left(3\right)}^{n-1}$

$\Rightarrow (3^8\times 3^{-n+1})={\left(3\right)}^{n-1}\Rightarrow {\left(3\right)}^{8-n+1}={\left(3\right)}^{n-1}\Rightarrow {\left(3\right)}^{-n+9}={\left(3\right)}^{n-1}\Rightarrow -n+9=n-1$

$\Rightarrow n+n=9+1\Rightarrow 2n=10\Rightarrow n=\frac{10}{2}\Rightarrow n=5$

Hence, $5$th term of the given sequence are equal.