$$a{r^4} = 2$$

$$a \times ar \times a{r^2} \times a{r^3} \times a{r^4} \times a{r^5} \times a{r^6} \times a{r^7} \times a{r^8}$$

$$ = {a^9}{r^{36}} = {\left( {a{r^4}} \right)^9} = {2^9} = 512$$

The product is $$p = {2^{1/4}}{.2^{2/8}}{.2^{3/16}}........$$

$$ = {2^{1/4 + 2/8 + 3/16 + .......\infty }}$$

Now let

$$S = {1 \over 4} + {2 \over 8} + {3 \over {16}} + .......\infty \,\,\,\,........\left( 1 \right)$$

$${1 \over 2}S = {1 \over 8} + {2 \over {16}} + .......\infty \,\,\,\,........\left( 2 \right)$$

Subtracting $$(2)$$ from $$(1)$$

$$ \Rightarrow {1 \over 2}S = {1 \over 4} + {1 \over 8} + {1 \over {16}} + .......\infty $$

or $${1 \over 2}S = {{1/4} \over {1 - 1/2}} = {1 \over 2} \Rightarrow S = 1$$

$$\therefore$$ $$P = {2^S} = 2$$

$$l = A{R^{p - 1}}$$

$$ \Rightarrow \log 1 = \log A + \left( {p - 1} \right)\log R$$

$$m = A{R^{q - 1}}$$

$$ \Rightarrow \log m = \log A + \left( {q - 1} \right)\log R$$

$$n = A{R^{r - 1}}$$

$$ \Rightarrow \log n = \log A + \left( {r - 1} \right)\log R$$

Now, $$\left| {\matrix{ {\log l} & p & 1 \cr {\log m} & q & 1 \cr {\log n} & r & 1 \cr } } \right|$$

$$ = \left| {\matrix{ {\log A + \left( {p - 1} \right)\log R} & p & 1 \cr {\log A + \left( {q - 1} \right)\log R} & q & 1 \cr {\log A + \left( {r - 1} \right)\log R} & r & 1 \cr } } \right|$$

Operating $${C_1} - \left( {\log R} \right){C_2} + \left( {\log R - \log A} \right){C_3}$$

$$ = \left| {\matrix{ 0 & p & 1 \cr 0 & q & 1 \cr 0 & r & 1 \cr } } \right| = 0$$

$$1,\,{\log _9}\left( {{3^{1 - x}} + 2} \right),{\log _3}\left( {{{4.3}^x} - 1} \right)$$ are in $$A.P.$$

$$ \Rightarrow 2{\log _9}\left( {{3^{1 - x}} + 2} \right)$$

$$\,\,\,\,\,\,\,\,\,$$ $$ = 1 + {\log _3}\left( {{{4.3}^x} - 1} \right)$$

$$ \Rightarrow {\log _3}\left( {{3^{1 - x}} + 2} \right)$$

$$\,\,\,\,\,\,\,\,\,$$ $$ = {\log _3}3 + {\log _3}\left( {{{4.3}^x} - 1} \right)$$

$$ \Rightarrow {\log _3}\left( {{3^{1 - x}} + 2} \right)$$

$$\,\,\,\,\,\,\,\,\,$$ $$ = {\log _3}\left[ {3\left( {{{4.3}^x} - 1} \right)} \right]$$

$$ \Rightarrow {3^{1 - x}} + 2 = 3\,\left( {{{4.3}^x} - 1} \right)$$

$$ \Rightarrow {3.3^{ - x}} + 2 = {12.3^x} - 3.$$

Put $${3^x} = t$$

$$ \Rightarrow {3 \over t} + 2 = 12t - 3$$

or $$12{t^2} - 5t - 3 = 0;$$

Hence $$t = - {1 \over 3},{3 \over 4} \Rightarrow {3^x} = {3 \over 4}$$

(as $${3^x}\,\, \ne \,\, - ve$$ )

$$ \Rightarrow x = {\log _3}\left( {{3 \over 4}} \right)$$

or $$x = {\log _3}3 - {\log _3}4$$

$$ \Rightarrow x = 1 - {\log _3}4$$