Expert: Paul Klarreich - 2/14/2006

Question
Hi i would like to know if i knew that Sum of the first 6 numbers in a geometric sequence is 252 and the sum of the first two numbers in the same sequence is 12 then how would i find out what the first four numbers are in the sequence
something to do with this formula:
Sn=a1-a1r^n
    -------------
        1-r

S6=252
S2=12
S4=?
A1=?
A2=?
A3=?
A4=?

Answer
Hi, Aqsa,

Subject:  Series
Question:  Hi.
i would like to know:

if i knew that:
Sum of the first 6 numbers in a geometric sequence is 252 and the sum of the first two numbers in the same sequence is 12 then

how would i find out what the first four numbers are in the sequence.  It has something to do with this formula:
Sn=  a1-a1r^n
   -------------
      1-r

------------------------------------------
Yes, that is the correct formula.  To determine what the series contains, you need two things:

The first term, usually denoted A1. (But I'll write 'a' for simplicity.)
The common ratio of consecutive terms, usually denoted r.

To find two variables, you need two facts, and you have them:

S2=12
S6=252

STANDARD WARNING: THE MATERIAL BELOW CONTAINS FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.

I'll use the formula in this format: (same as yours, of course)

    a(1 - r^n)
Sn = -----------
      1 - r

    a(1 - r^2)
12 = -----------
       1 - r

     a(1 - r^6)
252 = ------------
       1 - r

Now all you have to do is some algebra to find a and r.  Try dividing your S6 by your S2.  You will find the a's and the (1-r)'s canceling.

252   1 - r^6
--- = -------
12    1 - r^2

Now 1-r^2 is a factor of the top.  
1 - r^6 = (1 - r^2)(1 + r^2 + r^4)
So that equation reduces to:

21 = 1 + r^2 + r^4

r^4 + r^2 - 20 = 0, which you factor like a quadratic:

(r^2 + 5)(r^2 - 4) = 0,  which gives two real solutions:

r = 2  and r = -2.  Do both of these lead to legitimate series?

(1) Use r = 2 in the formula for S2 = a(1 + r)

12 = a(1 + 2)  
3a = 12
a = 4,

so the series is:

4,8,16,32,64,128 which adds to 252 OK.  What about r = -2?

(2) Use r = -2 in the formula for S2 = a(1 + r)

12 = a(1 - 2)
-1a = 12
a = -12

so the series is:

-12, 24, -48, 96, -192, 384.  Does that add up to 252?  Sure does.  Nothing wrong with this solution.  So there are actually two legitimate solutions to this problem.

How about that?