Chapter 12.2, Problem 13CFU

To determine

To calculate:

The geometric sequence which has two geometric means between $1$ and $27$

Answer to Problem 13CFU

The geometric sequence is $1,10.9774,17.2147,27$ .

Explanation of Solution

Given information:

  $\begin{array}{l}{a}_1=1\\ a_4=27\end{array}$

Formula used:

The $n^{th}$ term of the geometric sequence is given by the following expression:

  $a_n=a_1{r}^{n-1}$

Calculation:

The $1^{st}$ term of the geometric sequence is calculated by using the $n^{th}$ term:

  $\begin{array}{l}{a}_4=a_1{r}^{4-1}\\ 27=7r^3\\ r^3=\frac{27}{7}\\ r^3=3.8571\\ r=1.5682\end{array}$

Now, $2^{nd}$ and $3^{rd}$ term is calculated as:

  $\begin{array}{l}{a}_2=a_1{r}^{2-1}=\left(7\right){\left(1.5682\right)}^{2-1}=\left(7\right){\left(1.5682\right)}^1=10.9774\\ a_3=a_1{r}^{3-1}=\left(7\right){\left(1.5682\right)}^{3-1}=\left(7\right){\left(1.5682\right)}^2=17.2147\end{array}$

So, geometric sequence is written as:

  $1,10.9774,17.2147,27$