Question 1037774
For the first question:

{{{g[1]+g[2] = 8}}} and {{{g[3]+g[4] = 72}}} ==> {{{g[1]+g[1]*r = g[1]*(1+r) = 8}}}, and {{{g[3]+g[4] = g[1]*r^2 + g[1]*r^3 = g[1]*r^2(1+r) = 72}}}.

Dividing the second equation by the first equation, we get {{{r^2 = 9}}}, which means r = 3.  (r = -3 is not acceptable, as the ratio should be positive.)

The sequence is thus 2, 6, 18, 54.


For the second question:
{{{g[1]+g[1]*r + g[1]*r^2 = 42 }}} and {{{g[1]*g[1]r*g[1]r^2 = 512}}}

The second equation gives {{{(g[1]*r)^3 = 512}}}, which implies that  {{{g[1]*r = 8}}}, after taking cube roots of both sides.

The first equation {{{g[1]+g[1]*r + g[1]*r^2 = 42 }}} is equivalent to 
{{{g[1]*(1+r+r^2) = 42}}}.  Dividing this equation with {{{g[1]*r = 8}}}, we get
{{{(1+r+r^2)/r = 21/4}}}

==> {{{4r^2+4r+4 = 21r}}} <==>  {{{4r^2 - 17r+4 = 0}}} <==> r = 1/4 or 4.

If r = 1/4, the sequence is 32, 8, 2.
If r = 4, the sequence is 2, 8, 32.