Chapter 11.3, Problem 36PPS

To determine

Find three geometric means between 640 and 2.5.

Answer to Problem 36PPS

Three geometric means are: $160,40,10$

Explanation of Solution

Given:

First and last terms are 640 and 2.5

Concept Used:

Since there are 3 terms between the first and last term, there are 3+2= 5 total terms, so n = 5

Now find the common difference.

Calculation:

Now find the common difference.

  $\begin{array}{l}{a}_1=640\text{and}{a}_5=2.5;\text{find}r\\ a_5=a_1·r^{5-1}=a_1·r^4\\ 2.5=810·r^4\\ \frac{2.5}{640}=\frac{640·r^4}{640}\\ $ r^4=\frac{1}{256}={\left(\frac{1}{4}\right)}^4$r=\frac{1}{4}\end{array}$

  $\begin{array}{l}{a}_1=640;r=\frac{1}{4},\text{Find}{a}_2,a_3,a_4,\\ a_2=a_1·r=640·\frac{1}{4}=160\\ a_3=a_2·r=160·\frac{1}{4}=40\\ a_4=a_3·r=40·\frac{1}{4}=10\end{array}$

Three geometric means are: $160,40,10$

Thus, three geometric means are: $160,40,10$