Question 988607
if the fourth term {{{a[4]}}}  exceeds the third term {{{a[3]}}}  by {{{54}}}, we have

{{{a[4]-a[3] =54}}}....eq.1

 if the sum of the second {{{a[2]}}}and third term {{{a[3]}}} is {{{36}}}, we have

{{{a[2]+a[3]=36}}}....eq.2

By using {{{a[n]=a[1]*r^(n-1)}}} we have:


{{{a[1]*r^3-a[1]*r^2=54}}}....eq.1

{{{a[1]*r+a[1]*r^2=36}}}....eq.2
---------------------------------------------------add eq.2 and eq.1


{{{a[1]*r^3-a[1]*r^2+a[1]*r+a[1]*r^2=54+36}}}

{{{a[1]*r^3+a[1]*r=90}}}

{{{a[1](r^3+r)=90}}}

{{{a[1]=90/(r^3+r)}}}.............substitute in eq.2


{{{90/(r^3+r)*r+90/(r^3+r)*r^2=36}}}....eq.2

{{{90r/(r^3+r)+90r^2/(r^3+r)=36}}}

{{{(90r+190r^2)=36(r^3+r)}}}

{{{90r(1+r)=36r(r^2+1)}}}......divide by {{{18}}}

{{{5(1+r)=2(r^2+1)}}}

{{{5+5r=2r^2+2}}}

{{{2r^2+2-5r-5=0}}}

{{{2r^2-5r-3=0}}}

{{{(r-3) (2 r+1) = 0}}}

=> one solution will be: {{{(r-3) = 0}}}=> {{{highlight(r=3)}}}
=> another solution will be: {{{2r+1=0}}}=>{{{2r=-1}}}=>{{{highlight(r=-1/2)}}}

now we can find first term:

if   {{{highlight(r=3)}}}

{{{a[1]*(3)+a[1]*(3)^2=36}}}....eq.2

{{{3a[1]+9a[1]=36}}}
{{{12a[1]=36}}}
{{{highlight(a[1]=3)}}}


if {{{highlight(r=-1/2)}}}:

{{{a[1]*(-1/2)+a[1]*(-1/2)^2=36}}}....eq.2

{{{a[1]*(-1/2)+a[1]*(1/4)=36}}}......multiply by {{{4}}}

{{{-2a[1]+a[1]=144}}}
{{{-a[1]=144}}}
{{{highlight(a[1]= -144)}}}


so, there are two solutions:
1. {{{highlight(a[1]= 3)}}} and {{{highlight(r=3)}}}
2.{{{highlight(a[1]= -144)}}} and {{{highlight(r=-1/2)}}}



then the second term is:

if {{{a[1]= 3}}} and {{{r=3}}}:

{{{a[2]=3*3}}}
{{{a[2]=9}}}

third term is

{{{a[3]=3*3^2}}}
{{{a[3]=27}}}

fourth term
{{{a[4]=3*3^3}}}
{{{a[4]=3*27}}}
{{{a[4]=81}}}

the terms of sequence are:=>{{{3,9,27,81}}}



if {{{a=-144}}},{{{r=-1/2}}}

{{{a[2]= -144*(-1/2)^1}}}

{{{a[2]=-144/-2}}}

{{{a[2]=72}}}.

third term is

{{{a[3]=-144*(-1/2)^2}}}

{{{a[3]= -144(1/4)}}}

{{{a[3]= -36}}}

and fourth term is

{{{a[4]= -144*(-1/2)^3}}} 

{{{a[4]= -144(-1/8)}}}

{{{a[4]=18}}}

the terms of sequence are:=>{{{-144,72,-36,18}}}


now, check given data:

for =>{{{3,9,27,81}}}

{{{a[4]=a[3] +54}}}....eq.1
{{{81=27 +54}}}
{{{81=81}}}

{{{a[2]+a[3]=36}}}....eq.2
{{{9+27=36}}}
{{{36=36}}}


for =>{{{-144,72,-36,18}}}

{{{a[4]=a[3] +54}}}....eq.1
{{{18=-36 +54}}}
{{{18=18}}}

{{{a[2]+a[3]=36}}}....eq.2
{{{72+(-36)=36}}}
{{{36=36}}}