$$n$$^th term of the given series

$$ = {T_n} = {\left( {n - 1} \right)^2} + \left( {n - 1} \right)n + {n^2}$$

$$ = {{\left( {{{\left( {n - 1} \right)}^3} - {n^3}} \right)} \over {\left( {n - 1} \right) - n}}$$

$$ = {n^3} - {\left( {n - 1} \right)^3}$$

$$ \Rightarrow {S_n} = \sum\limits_{k = 1}^n {\left[ {{k^3} - {{\left( {k - 1} \right)}^3}} \right]} $$

$$ \Rightarrow 8000 = {n^3}$$

$$ \Rightarrow n = 20\,\,$$ which is a natural number.

Now, put $$n = 1,2,3,.....20$$

$${T_1} = {1^3} - {0^3}$$

$${T_2} = {2^3} - {1^3}$$

.

.

.

$${T_{20}} = {20^3} - {19^3}$$

Now, $${T_1} + {T_2} + ..... + {T_{20}} = {S_{20}}$$

$$ \Rightarrow {S_{20}} = {20^3} - {0^3} = 8000$$

Hence, both the given statements are true and statement $$2$$ supports statement $$1.$$

Let required number of months $$=n$$

$$\therefore$$ $$200 \times 3 + \left( {240 + 280 + 320 + ...} \right.$$

$$\left. {\,\,\,\,\,\,\,\,\,\,\,\, + {{\left( {n - 3} \right)}^{th}}\,term} \right) = 11040$$

$$ \Rightarrow {{n - 3} \over 2}\left[ {2 \times 240 + \left( {n - 4} \right) \times 40} \right]$$

$$\,\,\,\,\,\,\,\,\,\,\,\, = 11040 - 600$$

$$ \Rightarrow \left( {n - 3} \right)\left[ {240 + 20n - 80} \right] = 10440$$

$$ \Rightarrow \left( {n - 3} \right)\left( {20n + 160} \right) = 10440$$

$$ \Rightarrow \left( {n - 3} \right)\left( {n + 8} \right) = 522$$

$$ \Rightarrow {n^2} + 5n - 546 = 0$$

$$ \Rightarrow \left( {n + 26} \right)\left( {n - 21} \right) = 0$$

$$\therefore$$ $$n = 21$$

Till $$10$$^th minute number of counted notes $$ = 1500$$

$$3000 = {n \over 2}\left[ {2 \times 148 + \left( {n - 1} \right)\left( { - 2} \right)} \right]$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = n\left[ {148 - n + 1} \right]$$

$$ \Rightarrow $$$${n^2} - 149n + 3000 = 0$$

$$ \Rightarrow n = 125,24$$

But $$n=125$$ is not possible

$$\therefore$$ total time $$ = 24 + 10 = 34$$ minutes.

We have

$$S = 1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + ........\infty \,\,\,\,\,...\left( 1 \right)$$

Multiplying both sides by $${1 \over 3}$$ we get

$${1 \over 3}S = {1 \over 3} + {2 \over {{3^2}}} + {6 \over {{3^3}}} + {{10} \over {{3^4}}} + .......\,\,\,\,\,...\left( 2 \right)$$

Subtracting eqn. $$(2)$$ from eqn. $$(1)$$ we get

$${2 \over 3}S = 1 + {1 \over 3} + {4 \over {{3^2}}} + {4 \over {{3^3}}} + {4 \over {{3^4}}} + .....\infty $$

$$ \Rightarrow {2 \over 3}S = {4 \over 3} + {4 \over {{3^2}}} + {4 \over {{3^3}}} + {4 \over {{3^4}}} + .....\infty $$

$$ \Rightarrow {2 \over 3}S = {{{4 \over 3}} \over {1 - {1 \over 3}}} = {4 \over 3} \times {3 \over 2}$$

$$ \Rightarrow S - 3$$