SOLUTION: For a geometric series, S2 = 20 and S3= 65. Find the first 3 terms.

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Question 1120323: For a geometric series, S2 = 20 and S3= 65. Find the first 3 terms.
Found 3 solutions by solver91311, ikleyn, greenestamps:
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!










So

and





First three terms are 5, 15, and 45


John

My calculator said it, I believe it, that settles it
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.
From the condition, we have

 =  =  = a*(1+r),              (1)

 =  =  = .    (2)


Divide (1) by (2). You will get

 =  =  = .

or

13(1+r) = 4*(1+r+r^2),

13 + 13r = 4 + 4r + 4r^2

4r^2 -9r - 9 = 0

 =  = ;

 =  = 3;    =  =  = .



Subtract (1) from (2). You will get 

 =  -  = 65 - 20 = 45 = .     (*)



Thus we should consider TWO opportunities:



    1.  If r = 3,  then according to (*)   = 45 =  = 9a,

        which implies  a = 5.


        Then   = 5,   = 3*5 = 15  and   = 3*15 = 45.

        Then   =  = 5 + 15 = 20  and   = 5 + 15 + 45 = 65    ! Correct !.


    2.  If r= , then according (*)  a =  = 5*16 = 80.  

        Then   =  = -60  and   =  = 45.

        In this way,   =  = 80 - 60 = 20  

                and    =  = 80 - 60 + 45 = 65.

        So, this opportunity does work, too.


Answer.  There are  TWO solutions.

         One solution is  a= 5, r= 3 and the three terms are  5, 15, 45.

         Another solution is  a= 80,  r=   and the three terms are  80, -60 and 45.


Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The first tutor found the solution with a positive constant ratio between terms; it can be found using a little educated trial and error and some mental arithmetic.

Tutor @ikleyn showed there are two solutions.

Here is a different approach to find both solutions.

Since S2 = 20 and S3 = 65, we know T3 = 45. So



Then







And we have what we need to find both solutions, one with r = -3/4 and one with r = 3.

T3 = 45 and r = 3 gives us the first three terms as 5, 15, and 45;
T3 = 45 and r = -3/4 gives us the first three terms as 80, -60, and 45.
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