If the sum of first 11 terms of an A.P.,
a₁, a₂, a₃, .... is 0 (a $$ \ne $$ 0), then the sum of the A.P.,
a₁ , a₃ , a₅ ,....., a₂₃ is ka₁ , where k is equal to :

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 :