Question 529087
For the sequence to be geometric the ratio between consecutive terms must be the same.
{{{(x+1)/(x-3)=(2x+8)/(x+1)}}}
You could use that definition to find your {{{x}}}.
Or you could use the fact that in a geometric sequence each middle term is the geometic mean of the neighboring terms. If a, b, and c are consecutive terms in a geometric sequence that means
{{{b=sqrt(ac)}}} or {{{b^2=ac}}}
Either way, you end up with
{{{(x+1)^2=(2x+8)(x-3)}}}
which simplifies to
{{{x^2=25}}}
so the solutions are
{{{x=5}}} and {{{x=-5}}}
That makes the first three terms -8, -4, and -2
and you should get the ratio and sum from that easily.
Without even using the formula for sum of a geometric sequence I realize that
-8+(-4)+(-2)+(-1)+(-1/2)+ ... gets closer and closer to -16, and the difference is always equal to that shrinking last term.