Let S be the sum of the first 9 terms of the series :
{x + k$$a$$} + {x² + (k + 2)$$a$$} + {x³ + (k + 4)$$a$$}
+ {x⁴ + (k + 6)$$a$$} + .... where a $$ \ne $$ 0 and x $$ \ne $$ 1.

If S = $${{{x^{10}} - x + 45a\left( {x - 1} \right)} \over {x - 1}}$$, then k is equal to :

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 :