Chapter EP, Problem 11.3.10EP

To determine

To find:The geometric means for the given sequence..

Answer to Problem 11.3.10EP

The geometric means for the sequence are $1,\pm 3,9,\pm 27,81$.

Explanation of Solution

Given: The given sequence is $1,\_\_\_,\_\_\_,\_\_\_,81$ .

The $n^{th}$ term of a geometric sequence is given by,

$a_n=a_1{r}^{n-1}$ ...... (1)

The first term of the sequence is,

$a_1=1$

Substitute the values equation (1).

$\begin{array}{c}{a}_5=1{\left(r\right)}^{5-1}\\ 81=r^4\\ r^4=81\\ r=\pm 3\end{array}$

The value of $a_2$ is calculated as,

$\begin{array}{c}{a}_2=1{\left(\pm 3\right)}^{2-1}\\ =1{\left(\pm 3\right)}^1\\ =\pm 3\end{array}$

The value of $a_3$ is calculated as,

$\begin{array}{c}{a}_3=1{\left(\pm 3\right)}^{3-1}\\ =1{\left(\pm 3\right)}^2\\ =9\end{array}$

The value of $a_4$ is calculated as,

$\begin{array}{c}{a}_4=1{\left(\pm 3\right)}^{4-1}\\ =1{\left(\pm 3\right)}^3\\ =\pm 27\end{array}$

Therefore, the geometric means for the sequence are $1,\pm 3,9,\pm 27,81$ .