If the value of

$${\left( {1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + ....upto\,\infty } \right)^{{{\log }_{(0.25)}}\left( {{1 \over 3} + {1 \over {{3^2}}} + {1 \over {{3^3}}} + ....upto\,\infty } \right)}}$$

is $$l$$, then $$l$$² is equal to _______________.

The sum of all the elements in the set {n$$\in$$ {1, 2, ....., 100} | H.C.F. of n and 2040 is 1} is equal to _____________.

For k $$\in$$ N, let $${1 \over {\alpha (\alpha + 1)(\alpha + 2).........(\alpha + 20)}} = \sum\limits_{K = 0}^{20} {{{{A_k}} \over {\alpha + k}}} $$, where $$\alpha > 0$$. Then the value of $$100{\left( {{{{A_{14}} + {A_{15}}} \over {{A_{13}}}}} \right)^2}$$ is equal to _____________.

Let $$\left\{ {{a_n}} \right\}_{n = 1}^\infty $$ be a sequence such that a₁ = 1, a₂ = 1 and $${a_{n + 2}} = 2{a_{n + 1}} + {a_n}$$ for all n $$\ge$$ 1. Then the value of $$47\sum\limits_{n = 1}^\infty {{{{a_n}} \over {{2^{3n}}}}} $$ is equal to ______________.