Chapter 11, Problem 32SGR

To determine

Find two geometric means between 6 and 162.

Answer to Problem 32SGR

  $6,\underset{\_}{18},\underset{\_}{54},162$

Explanation of Solution

Given:

Find two geometric means between 6 and 162.

Concept Used:

Since there are 2 terms between the first and last term, there are 2 + 2 = 4 total terms, so n = 4

Given $a_1=6\text{and}{a}_4=162.$

Formula for nth term of geometric series is $a_n=a_1·r^{n-1}$

Calculation:

Formula for nth term of geometric series is $a_n=a_1·r^{n-1}$

  $\begin{array}{l}{a}_1=6\text{and}{a}_4=162.\\ a_n=a_1·r^{n-1}\\ a_4=a_1·r^{4-1}=6·r^{4-1}\\ 162=6·r^3\Rightarrow r^3=27=3^3\\ r=3\end{array}$

The two geometric means are: $6·3=18\text{and}18·3=54$

The two geometric means are 18 and 54

  $6,\underset{\_}{18},\underset{\_}{54},162$

Thus, the two geometric means are 18 and 54