Question 696784
-9/2 + 3/2 - 1/2 + 1/6 - .......... + 1/39366
<pre>
r = {{{3/2}}}÷{{{-9/2}}} = {{{3/2}}}×{{{-2/9}}} = {{{(-6)/18}}} = {{{-1/3}}}

a<sub>n</sub> = a<sub>1</sub>r<sup>n-1</sup> = {{{(-9/2)}}}×{{{(-1/3)^(n-1)}}} = {{{(-9/2)}}}×{{{(-1/3)^(n-1)}}}

{{{(-9/2)}}}×{{{(-1/3)^(n-1)}}} = {{{1/39366}}}

Multiply both sides by {{{-2/9}}}

{{{-2/9}}}×{{{(-9/2)}}}×{{{(-1/3)^(n-1)}}} = {{{-2/9}}}×{{{1/39366}}}

             {{{(-1/3)^(n-1)}}} = {{{-2/9}}}×{{{1/39366}}}

             {{{(-1/3)^(n-1)}}} = {{{-2/(9*39366)}}}

             {{{(-1)^(n-1)(1/3)^(n-1)}}} = {{{-1/(9*19683)}}}

             {{{(-1)^(n-1)(1/3)^(n-1)}}} = {{{-1/177147}}}

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

             {{{(1/3)^(n-1)}}} = {{{1/177147}}} 

             {{{1/3^(n-1)}}} = {{{1/177147}}} 

taking reciprocals of both sides:

             3<sup>n-1</sup> = 177147

Taking the natural log of both sides

           ln(3<sup>n-1</sup>) = ln(177147)
           (n-1)ln(3) = ln(177147)
                  n-1 = {{{ln(177147)/ln(3)}}}
                  n-1 = 11
                    n = 12

So there are 12 terms:

Summation  = {{{sum((-9/2)(-1/3)^(n-1),1,12)}}}

S<sub>n</sub> = {{{(a[1](1-r^n))/(1-r)}}} = {{{(-9/2)(1-(-1/3)^12)/(1-(-1/3))}}} = {{{(-9/2)(1-(-1/3)^12)/(1+1/3)}}} = {{{(-9/2)(1-(-1/3)^12)/(4/3)}}}


= {{{(-9/2)(1-(-1/3)^12)*(3/4)}}} = {{{(-9/2)(1-1/3^12)*(3/4)}}} = {{{(-27/8)(1-1/3^12)}}} = -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</pre>