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) About Me  (Show Source):
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So

and





First three terms are 5, 15, and 45


John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
From the condition, we have

S%5B2%5D = a%5B1%5D+%2B+a%5B2%5D = a+%2B+ar = a*(1+r),              (1)

S%5B3%5D = a%5B1%5D+%2B+a%5B2%5D+%2B+a%5B3%5D = a%2Bar%2Bar%5E2 = a%2A%281%2Br%2Br%5E2%29.    (2)


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

%281%2Br%29%2F%281%2Br%2Br%5E2%29 = S%5B2%5D%2FS%5B3%5D = 20%2F65 = 4%2F13.

or

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

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

4r^2 -9r - 9 = 0

r%5B1%2C2%5D = %289+%2B-+sqrt%289%5E2+%2B+4%2A4%2A9%29%29%2F%282%2A4%29 = %289+%2B-+15%29%2F8;

r%5B1%5D = %289+%2B+15%29%2F8 = 3;   r%5B2%5D = %289+-+15%29%2F8 = -6%2F8 = -3%2F4.



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

a%5B3%5D = S%5B3%5D - S%5B2%5D = 65 - 20 = 45 = a%2Ar%5E2.     (*)



Thus we should consider TWO opportunities:



    1.  If r = 3,  then according to (*)  a%5B3%5D = 45 = a%2A3%5E2 = 9a,

        which implies  a = 5.


        Then  a%5B1%5D = 5,  a%5B2%5D = 3*5 = 15  and  a%5B3%5D = 3*15 = 45.

        Then  S%5B2%5D = a%5B1%5D%2Ba%5B2%5D = 5 + 15 = 20  and  S%5B3%5D = 5 + 15 + 45 = 65    ! Correct !.


    2.  If r= -3%2F4, then according (*)  a = 45%2F%28%289%2F16%29%29 = 5*16 = 80.  

        Then  a%5B2%5D = 80%2A%28-3%2F4%29 = -60  and  a%5B3%5D = -60%2A%28-3%2F4%29 = 45.

        In this way,  S%5B2%5D = a%5B1%5D%2Ba%5B2%5D = 80 - 60 = 20  

                and   S%5B3%5D = a%5B1%5D+%2B+a%5B2%5D+%2B+a%5B3%5D = 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= -3%2F4  and the three terms are  80, -60 and 45.


Answer by greenestamps(13200) About Me  (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
S2+=+a%2Bar+=+20
T3+=+ar%5E2+=+45

Then

%28a%2Bar%29%2F%28ar%5E2%29+=+20%2F45+=+4%2F9
%281%2Br%29%2Fr%5E2+=+4%2F9
4r%5E2+=+9r%2B9
4r%5E2-9r-9+=+0
%284r%2B3%29%28r-3%29+=+0

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.