Sum of the first n terms of the series $${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$$ is equal to

If $$a,\,b,\,c$$ are in GP., then the equations $$\,\,\alpha {x^2} + 2bx + c = 0$$ and $$d{x^2} + 2ex + f = 0$$ have a common root if $${d \over a},\,{e \over b},{f \over c}$$ are in ________.

The rational number, which equals the number $$2\overline {357} $$ with recurring decimal is

The third term of a geometric progression is 4. The product of the first five terms is