If for x, y $$\in$$ R, x > 0, y = log₁₀x + log₁₀x^1/3 + log₁₀x^1/9 + ...... upto $$\infty$$ terms

and $${{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ..... + 3y}} = {4 \over {{{\log }_{10}}x}}$$, then the ordered pair (x, y) is equal to :

The sum of the series

$${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$$ when x = 2 is :

If the sum of an infinite GP a, ar, ar², ar³, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar², ar⁴, ar⁶, ....... is :

Let S_n be the sum of the first n terms of an arithmetic progression. If S_3n = 3S_2n, then the value of $${{{S_{4n}}} \over {{S_{2n}}}}$$ is :