Question 988607
{{{r}}}= the common ratio.
{{{a[2]}}}= the second term.
{{{a[3]=a[2]*r}}}= the third term.
{{{a[3]=a[3]*r=a[2]*r^2}}}= the fourth term.
The fourth term of a geometric progression exceeds the third term by 54 translates as
{{{a[4]-a[3]=54}}}<-->{{{a[2]*r^2-a[2]*r=54}}}<-->{{{a[2]*(r^2-r)=54}}} .
The sum of the second and third term is 36 translates as
{{{a[2]+a[3]=36}}}<-->{{{a[2]+a[2]*r=36}}}<-->{{{a[2]*(1+r)=36}}} .
The ratio both equations tells us that
{{{(a[2]*(r^2-r))/(a[2]*(1+r))=54/36}}}-->{{{(r^2-r)/(1+r)=3/2}}}-->{{{2(r^2-r)=3(1+r)}}}-->{{{2r^2-2r)=3+3r}}}}-->{{{2r^2-5r-3=0}}} .
The equation {{{2r^2-5r-3=0}}} is a quadratic equation.
As is true for all quadratic equations, it can be solved by "completing the square, or by using the quadratic formula.
This particular quadratic equation can also be solved by factoring:
{{{2r^2-5r-3=0}}}--->{{{2r^2-6r+r-3=0}}}--->{{{2r(r-3)+r-3=0}}}}--->{{{(2r+1)(r-3)=0}}}--->{{{system(highlight(r=3),"or",r=-1/2)}}} .
If the common ratio is positive, it must be {{{highlight(r=3)}}} .