If p, q, r are in AP and are positive, the roots of the quadratic equation px² + qx + r = 0 are all real for

If one GM, g and two AM's p and q are inserted between two numbers a and b, then (2p $$-$$ q) (p $$-$$ 2q) is equal to

Given that x, y, and z are three consecutive positive integers and x $$-$$ z + 2 = 0, what is the value of $${1 \over 2}{\log _e}x + {1 \over 2}{\log _e}z + {1 \over {2xz + 1}} + {1 \over 3}{\left( {{1 \over {2xz + 1}}} \right)^3} + ...$$?

The value of the sum $$\sum\limits_{k = 1}^\infty {\sum\limits_{n = 1}^\infty {{k \over {{2^{n + k}}}}} } $$ is