The product $${2^{{1 \over 4}}}{.4^{{1 \over {16}}}}{.8^{{1 \over {48}}}}{.16^{{1 \over {128}}}}$$ ... to $$\infty $$ is equal to :

If the 10^th term of an A.P. is $${1 \over {20}}$$ and its 20^th term is $${1 \over {10}}$$, then the sum of its first 200 terms is

Let ƒ : R $$ \to $$ R be such that for all x $$ \in $$ R
(2^1+x + 2^1–x), ƒ(x) and (3^x + 3^–x) are in A.P.,
then the minimum value of ƒ(x) is

Let $${a_1}$$ , $${a_2}$$ , $${a_3}$$ ,....... be a G.P. such that
$${a_1}$$ < 0, $${a_1}$$ + $${a_2}$$ = 4 and $${a_3}$$ + $${a_4}$$ = 16.
If $$\sum\limits_{i = 1}^9 {{a_i}} = 4\lambda $$, then $$\lambda $$ is equal to: