Putnam 1950 B5

5. Answer either (i) or (ii). (i) Given that the sequence whose $n$th term is $\left(s_{n}+2 s_{n+1}\right)$ converges, show that the sequence $\left\{s_{n}\right\}$ converges also. (page 306) (ii) A plane varies so that it includes a cone of constant volume equal to $\pi a^{3} / 3$ with the surface the equation of which in rectangular coordinates is $2 x y=z^{2}$. Find the equation of the envelope of the various positions of this plane.

State the result so that it applies to a general cone (that is, conic surface) of the second order.