Question 775436
The three numbers are in geometric progression, sp
{{{b/a=r}}}<-->{{{a=b/r}}} and
{{{c/b=r}}}<-->{{{c=br}}}
with {{{r}}}= common ratio of the progression.
Since a,b,c are three distinct numbers,
{{{r<>1}}} and {{{r<>-1}}}
 
{{{a+b+c=b/r+b+br=(b+br+br^2)/r=b(1+r+r^2)/r}}}
If {{{a+b+c=xb}}} {{{xb=b(1+r+r^2)/r}}} --> {{{x=(1+r+r^2)/r}}}
 
If {{{r<0}}} and {{{r<>-1}}},
{{{x+1=(1+r+r^2)/r+1=(1+2r+r^2)/r=(r+1)^2/r<0}}}
and {{{x+1<0}}} --> {{{highlight(x<-1)}}}
 
If {{{r>0}}} and {{{r<>1}}},
{{{x-3=(1+r+r^2)/r-3=(1-2r+r^2)/r=(r-1)^2/r>0}}}
and {{{x-3>0}}} --> {{{highlight(x>3)}}}