If |x| < 1, |y| < 1 and x $$ \ne $$ y, then the sum to infinity of the following series

(x + y) + (x²+xy+y²) + (x³+x²y + xy²+y³) + ....

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

Let a_n be the n^th term of a G.P. of positive terms.

$$\sum\limits_{n = 1}^{100} {{a_{2n + 1}} = 200} $$ and $$\sum\limits_{n = 1}^{100} {{a_{2n}} = 100} $$,

then $$\sum\limits_{n = 1}^{200} {{a_n}} $$ is equal to :

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