r = ÷ = × =  = 

an = a1rn-1 = × = ×

× = 

Multiply both sides by 

×× = ×

              = ×

              = 

              = 

              = 

n-1 must be an odd number since the right side is negative,
so (-1)n-1 = -1.   Therefore, dividing both sides
by -1,

              =  

              =  

taking reciprocals of both sides:

             3n-1 = 177147

Taking the natural log of both sides

           ln(3n-1) = ln(177147)
           (n-1)ln(3) = ln(177147)
                  n-1 = 
                  n-1 = 11
                    n = 12

So there are 12 terms:

Summation  = 

Sn =  =  =  = 


=  =  =  = -3.37499365   

The computer program I have is Liberty Basic of which you can download
a free trial version.   Here is the Liberty Basic program I wrote

A1 = -9/2
r = -1/3
for n = 1 to 12
An = A1*r^(n-1)
sum = sum+An
print "A(";n;") = ";An;", sum =";sum
next
 
Here is the output of that program:

A(1) = -4.5, sum =-4.5
A(2) = 1.5, sum =-3
A(3) = -0.5, sum =-3.5
A(4) = 0.16666667, sum =-3.33333333
A(5) = -0.55555556e-1, sum =-3.38888889
A(6) = 0.18518519e-1, sum =-3.37037037
A(7) = -0.61728395e-2, sum =-3.37654321
A(8) = 0.20576132e-2, sum =-3.3744856
A(9) = -0.68587106e-3, sum =-3.37517147
A(10) = 0.22862369e-3, sum =-3.37494284
A(11) = -0.76207895e-4, sum =-3.37501905
A(12) = 0.25402632e-4, sum =-3.37499365
  
Edwin