If the sum of the first 20 terms of the series
$${\log _{\left( {{7^{1/2}}} \right)}}x + {\log _{\left( {{7^{1/3}}} \right)}}x + {\log _{\left( {{7^{1/4}}} \right)}}x + ...$$ is 460,
then x is equal to :

If 2¹⁰ + 2⁹.3¹ + 2⁸ .3² +.....+ 2.3⁹ + 3¹⁰ = S - 2¹¹, then S is equal to :

If $${3^{2\sin 2\alpha - 1}}$$, 14 and $${3^{4 - 2\sin 2\alpha }}$$ are the first three terms of an A.P. for some $$\alpha $$, then the sixth terms of this A.P. is:

The minimum value of 2^sinx + 2^cosx is :