Chapter 11.3, Problem 25WE

To determine

To calculate: The geometric means between the given pairs.

Answer to Problem 25WE

The geometric means between the pairs are $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.},4^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.},8^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.},{\text{ and 16}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$ .

Explanation of Solution

Given information: first term, a = 1 and last term, $t_n$ = 2

With 4 geometric means between, since total number of terms are 6

Hence, $t_1=1,t_2,t_3,t_4,t_5,t_6=2$

Apply the formula to calculate the n^th term of geometric sequence to find the common ratio r .

Let,

  $\begin{array}{l}a=t_1\text{ and }{t}_n=t_6\\ \therefore n=6\end{array}$

Formula used: $t_n=a.r^{n-1}$

Calculation:

  $\begin{array}{l}\Rightarrow 2=\left(1\right)\times r^{6-1}=r^5\\ \Rightarrow r=2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\end{array}$

Apply the formula to calculate the middle 4 terms

Therefore, $t_n=a.r^{n-1}$

  $\therefore t_2=\left(1\right)\times {\left(2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)}^{2-1}=2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$

  $\text{Or,}{t}_3=\left(1\right)\times {\left(2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)}^{3-1}=2^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}=4^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$

  $\begin{array}{l}\text{Or,}{t}_4=\left(1\right)\times {\left(2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)}^{4-1}=2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}=8^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\\ \text{Or, }{t}_5=\left(1\right)\times {\left(2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\right)}^{5-1}=2^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}={16}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\end{array}$

Hence, geometric means between the pairs are $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.},4^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.},8^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.},{\text{ and 16}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$ .