Chapter 9.3, Problem 24PSD

(i)

To determine

To find: The Arithmetic, Geometric and Harmonic mean if $AD=2\text{ \& }BD=8$.

Answer to Problem 24PSD

  $AM=5,GM=4\text{ and }HM=\frac{16}{5}$

Explanation of Solution

Given information :

  $AD=2\text{ and }BD=8$

Formula used :

  $AM=\frac{a+b}{2},GM=\sqrt{ab} \text{and }HM=\frac{2ab}{a+b}$

Calculation :

  $AB=\frac{\text{2+8}}{\text{2}}=\frac{10}{2}=5$

  $GM=\sqrt{\text{2}\times \text{8}}=\sqrt{16}=4$

  $HM=\frac{\text{2}\times \text{2}\times \text{8}}{\text{2+8}}=\frac{32}{10}=\frac{16}{5}=3\frac{1}{5}$

(ii)

To determine

To find: The Arithmetic, Geometric and Harmonic mean if $AD=3\text{ \& }BD=12$.

Answer to Problem 24PSD

  $AM=7\frac{1}{2},GM=6\text{ and }HM=4\frac{4}{5}$

Explanation of Solution

Given information :

  $AD=3\text{ and }BD=12$

Formula used :

  $AM=\frac{a+b}{2},GM=\sqrt{ab} \text{and }HM=\frac{2ab}{a+b}$

Calculation :

  $AM=\frac{\text{3+12}}{\text{2}}=\frac{15}{2}=7\frac{1}{2}$

  $GM=\sqrt{3\times 12}=\sqrt{36}=6$

  $HM=\frac{\text{2}\times 3\times 12}{\text{3+12}}=\frac{72}{15}=\frac{24}{5}=4\frac{4}{5}$

(iii)

To determine

To find: The Arithmetic, Geometric and Harmonic mean if $AD=4\text{ \& }BD=25$.

Answer to Problem 24PSD

  $AM=14\frac{1}{2},GM=10\text{ and }HM=6\frac{26}{29}$

Explanation of Solution

Given information :

  $AD=4\text{ and }BD=25$

Formula used :

  $AM=\frac{a+b}{2},GM=\sqrt{ab} \text{and }HM=\frac{2ab}{a+b}$

Calculation :

  $AM=\frac{\text{4+25}}{\text{2}}=\frac{29}{2}=14\frac{1}{2}$

  $GM=\sqrt{4\times 25}=\sqrt{100}=10$

  $HM=\frac{\text{2}\times 4\times 25}{\text{4+25}}=\frac{200}{29}=6\frac{26}{29}$

iv

To determine

To prove: That $AM(\text{Arithmetic Mean})\ge GM(\text{Geometric Mean})$

Explanation of Solution

Given information :

  $AM=\frac{a+b}{2},GM=\sqrt{ab}$

Formula used :

  $AM=\frac{a+b}{2},GM=\sqrt{ab}$

Proof :

  $\begin{array}{l}{\left(a+b\right)}^2=a^2+2ab+b^2\ge 0\\ \Rightarrow {\left(a+b\right)}^2-2ab\ge 0\\ \Rightarrow {\left(a+b\right)}^2\ge 2ab\\ \Rightarrow \left(a+b\right)\ge \sqrt{2ab}\\ \Rightarrow \frac{a+b}{2}\ge \sqrt{2}\cdot \sqrt{ab}\\ \Rightarrow AM\ge GM\end{array}$