Question 732805
Geometric sequence must have a common ratio for each successive term.
{{{x/1=y/x}}}
{{{y=x^2}}}.


Arithmetic sequence must have a common difference for each successive term.
{{{x-1=-y-x}}}
{{{2x-1=-y}}}
{{{y=-2x+1}}}.


There occur two different formulas for y.  We equate them:
{{{x^2=-2x+1}}}
{{{x^2+2x-1=0}}} That proved!


Find x through either completing the square or, preferably, use solution to quadratic formula.  {{{x=(-2+sqrt(4-4*(-1)))/2}}} OR {{{x=(-2-sqrt(4-4*(-1)))/2}}},
{{{highlight(x=-1-sqrt(2))}}} OR {{{highlight(x=-1+sqrt(2))}}}, and you can continue for finding y.