Given,  b  is the mean proportion between  a  and  c.

$$\begin{gathered} \Rightarrow \frac{\mathrm{a}}{\mathrm{b}} = \frac{\mathrm{b}}{\mathrm{c}} = \mathrm{k} ( \mathrm{Say} ) \\ \Rightarrow \mathrm{a} = \mathrm{b} \mathrm{k}, \mathrm{b} = \mathrm{c} \mathrm{k} \\ \Rightarrow \mathrm{a} = ( \mathrm{c} \mathrm{k} ) \mathrm{k} = \mathrm{c} {\mathrm{k}}^2, \mathrm{b} = \mathrm{c} \mathrm{k} \\ \mathrm{L}.\mathrm{H}.\mathrm{S}. = \frac{{\mathrm{a}}^4 + {\mathrm{a}}^2 {\mathrm{b}}^2 + {\mathrm{b}}^4}{{\mathrm{b}}^4 + {\mathrm{b}}^2 {\mathrm{c}}^2 + {\mathrm{c}}^4} \\ = \frac{( \mathrm{c} {\mathrm{k}}^2 {)}^4 +( \mathrm{c} {\mathrm{k}}^2 {)}^2 ( \mathrm{c} \mathrm{k} {)}^2 + ( \mathrm{c} \mathrm{k} {)}^4}{( \mathrm{c} \mathrm{k} {)}^4 + ( \mathrm{c} \mathrm{k} {)}^2 {\mathrm{c}}^2 + {\mathrm{c}}^4} \end{gathered}$$

           $$\begin{gathered} = \frac{ {\mathrm{c}}^4 {\mathrm{k}}^8 + ( {\mathrm{c}}^2 {\mathrm{k}}^4 ) ( {\mathrm{c}}^2 {\mathrm{k}}^2 ) + {\mathrm{c}}^4 {\mathrm{k}}^4}{ {\mathrm{c}}^4 {\mathrm{k}}^4 + ( {\mathrm{c}}^2 {\mathrm{k}}^2 ) {\mathrm{c}}^2 + {\mathrm{c}}^4} \\ = \frac{ {\mathrm{c}}^4 {\mathrm{k}}^8 + {\mathrm{c}}^4 {\mathrm{k}}^6 + {\mathrm{c}}^4 {\mathrm{k}}^4}{ {\mathrm{c}}^4 {\mathrm{k}}^4 + {\mathrm{c}}^4 {\mathrm{k}}^2 + {\mathrm{c}}^4} \\ = \frac{{\mathrm{c}}^4 {\mathrm{k}}^4 \left( {\mathrm{k}}^4 + {\mathrm{k}}^2 + 1 \right)}{{\mathrm{c}}^4 \left( {\mathrm{k}}^4 + {\mathrm{k}}^2 + 1 \right)} \\ = {\mathrm{k}}^4 \end{gathered}$$

$$\begin{gathered} \mathrm{R}.\mathrm{H}.\mathrm{S}. = \frac{{\mathrm{a}}^2}{{\mathrm{c}}^2} \\ = \frac{( \mathrm{c} {\mathrm{k}}^2 {)}^2}{{\mathrm{c}}^2} \\ = \frac{ {\mathrm{c}}^2 {\mathrm{k}}^4}{{\mathrm{c}}^2} \\ = {\mathrm{k}}^4 \end{gathered}$$

Hence,   L.H.S.  =  R.H.S.

$$\begin{gathered} \mathrm{Let} \frac{\mathrm{x}}{\mathrm{a}} = \frac{\mathrm{y}}{\mathrm{b}} = \frac{\mathrm{z}}{\mathrm{c}} = \mathrm{k} \\ \Rightarrow \mathrm{x} = \mathrm{a} \mathrm{k}, \mathrm{y} = \mathrm{b} \mathrm{k}, \mathrm{z} = \mathrm{c} \mathrm{k} \\ \mathrm{L}.\mathrm{H}.\mathrm{S}. =\frac{{\mathrm{x}}^3}{{\mathrm{a}}^3} + \frac{{\mathrm{y}}^3}{{\mathrm{b}}^3} + \frac{{\mathrm{z}}^3}{{\mathrm{c}}^3} \\ = \frac{( \mathrm{a} \mathrm{k} {)}^3}{{\mathrm{a}}^3} + \frac{( \mathrm{b} \mathrm{k} {)}^3}{{\mathrm{b}}^3} + \frac{( \mathrm{c} \mathrm{k} {)}^3}{{\mathrm{c}}^3} \\ = \frac{ {\mathrm{a}}^3 {\mathrm{k}}^3}{{\mathrm{a}}^3} + \frac{ {\mathrm{b}}^3 {\mathrm{k}}^3}{{\mathrm{b}}^3} + \frac{{\mathrm{c}}^3 {\mathrm{k}}^3 }{{\mathrm{c}}^3} \\ = {\mathrm{k}}^3 + {\mathrm{k}}^3 + {\mathrm{k}}^3 \\ = 3 {\mathrm{k}}^3 \end{gathered}$$

$$\begin{gathered} \mathrm{R}.\mathrm{H}.\mathrm{S}. = \frac{3 \mathrm{x} \mathrm{y} \mathrm{z}}{\mathrm{a} \mathrm{b} \mathrm{c}} \\ = \frac{3 ( \mathrm{a} \mathrm{k} ) ( \mathrm{b} \mathrm{k} ) ( \mathrm{c} \mathrm{k} )}{\mathrm{a} \mathrm{b} \mathrm{c}} \\ = 3 {\mathrm{k}}^3 \\ = \mathrm{L}.\mathrm{H}.\mathrm{S}. \\ \Rightarrow \mathrm{L}.\mathrm{H}.\mathrm{S}. = \mathrm{R}.\mathrm{H}.\mathrm{S}. \\ \Rightarrow \frac{{\mathrm{x}}^3}{{\mathrm{a}}^3} + \frac{{\mathrm{y}}^3}{{\mathrm{b}}^3} + \frac{{\mathrm{z}}^3}{{\mathrm{c}}^3} = \frac{3 \mathrm{x} \mathrm{y} \mathrm{z}}{\mathrm{a} \mathrm{b} \mathrm{c}} \end{gathered}$$

We know,

Population after n years = Present Population $\times$ $(1 + \frac{\mathrm{r}}{100}{)}^{\mathrm{n}}$

As given Present population = 2,00,000

After first year, population = 2,00,000 $\times (1 + \frac{10}{100}{)}^1$

                                       = 2,00,000 $\times \frac{11}{10}$

                                       = 2,20,000

Population after two years = 2,20,000 $\times (1 + \frac{15}{100}{)}^2$

                                       = 2,53,000

Thus, the population after two years is 2,53,000

$( \mathrm{i} ) \frac{7 \mathrm{m} + 2 \mathrm{n}}{7 \mathrm{m} - 2 \mathrm{n}} = \frac{5}{3}$

By Componendo - Divinendo,  we get

$$\begin{gathered} \frac{7 \mathrm{m} + 2 \mathrm{n} + \left( 7 \mathrm{m} - 2 \mathrm{n} \right)}{7 \mathrm{m} + 2 \mathrm{n} - \left( 7 \mathrm{m} - 2 \mathrm{n} \right)} = \frac{5 + 3}{5 - 3} \\ \Rightarrow \frac{14 \mathrm{m}}{4 \mathrm{n}} = \frac{8}{2} \\ \Rightarrow \frac{7 \mathrm{m}}{2 \mathrm{n}} = \frac{4}{1} \\ \Rightarrow \frac{ \mathrm{m}}{ \mathrm{n}} = \frac{8}{7} \\ \Rightarrow \mathrm{m} : \mathrm{n} = 8 : 7 \end{gathered}$$

( ii ) $\frac{\mathrm{m}}{\mathrm{n}} = \frac{8}{7} \Rightarrow \frac{{\displaystyle {\mathrm{m}}^2}}{{\displaystyle {\mathrm{n}}^2}} = \frac{{\displaystyle 8^2}}{{\displaystyle 7^2}}$

Applying  Componendo - Divinendo,  we get

$$\begin{gathered} \Rightarrow \frac{{\mathrm{m}}^2 + {\mathrm{n}}^2}{{\mathrm{m}}^2 - {\mathrm{n}}^2} = \frac{8^2 + 7^2}{8^2 - 7^2} \\ \Rightarrow \frac{{\mathrm{m}}^2 + {\mathrm{n}}^2}{{\mathrm{m}}^2 - {\mathrm{n}}^2} = \frac{64 + 49}{64 - 49} \\ \Rightarrow \frac{{\mathrm{m}}^2 + {\mathrm{n}}^2}{{\mathrm{m}}^2 - {\mathrm{n}}^2} = \frac{113}{15} \end{gathered}$$

$$\begin{gathered} \frac{2 \mathrm{x} + \sqrt{ 4 {\mathrm{x}}^2 - 1}}{2 \mathrm{x} - \sqrt{ 4 {\mathrm{x}}^2 - 1}} = 4 \\ \Rightarrow \frac{{\displaystyle 2 \mathrm{x} + \sqrt{ 4 {\mathrm{x}}^2 - 1} + 2 \mathrm{x} - \sqrt{ 4 {\mathrm{x}}^2 - 1}}}{{\displaystyle 2 \mathrm{x} + \sqrt{ 4 {\mathrm{x}}^2 - 1} -( 2 \mathrm{x} - \sqrt{ 4 {\mathrm{x}}^2 - 1} )}} = \frac{4 + 1}{4 - 1} ( \mathrm{By} \mathrm{componendo} - \mathrm{dividendo} ) \\ \Rightarrow \frac{ 4 \mathrm{x}}{2 \sqrt{ 4 {\mathrm{x}}^2 - 1}} = \frac{5}{3} \\ \Rightarrow \frac{{\displaystyle 2 \mathrm{x}}}{{\displaystyle \sqrt{ 4 {\mathrm{x}}^2 - 1}}} = \frac{{\displaystyle 5}}{{\displaystyle 3}} \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{4 {\mathrm{x}}^2}{4 {\mathrm{x}}^2 - 1} = \frac{25}{9} ( \mathrm{Squaring} \mathrm{both} \mathrm{sides} ) \\ \Rightarrow \frac{4 {\mathrm{x}}^2 - 4 {\mathrm{x}}^2 + 1}{4 {\mathrm{x}}^2 - 1} = \frac{25 - 9}{9} ( \mathrm{By} \mathrm{dividendo} ) \\ \Rightarrow \frac{1}{4 {\mathrm{x}}^2 - 1} = \frac{16}{9} \\ \Rightarrow 9 = 64 {\mathrm{x}}^2 - 16 \\ \Rightarrow 64 {\mathrm{x}}^2 = 9 + 16 \\ = 25 \\ \Rightarrow {\mathrm{x}}^{2 } = \frac{25}{64} \\ \Rightarrow {\mathrm{x}}^{ } = \pm \frac{5}{8} \\ \Rightarrow {\mathrm{x}}^{ } = \frac{5}{8} ( \mathrm{x} \mathrm{is} \mathrm{positive} ) \end{gathered}$$