Geometric Sequences and Series

Date: 09/10/99 at 00:30:34
From: JC
Subject: Geometric Sequences and Series

I'm having a hard time coming up with formulas that explain geometric 
sequences and series. The following problem was given to me and we're 
supposed to do many other things with it, but without a formula I'm 
stuck.

The sequence starts out like this:

1. Consider the geometric sequence which begins -3072, 1536, -768

After a general formula is given to explain the sequence I can answer 
the problems associated with it:

   a. Find the 13th and 20th terms.
   b. Find the sum of the first 9 terms.

Date: 09/29/1999 at 13:05:06
From: Doctor Dwayne
Subject: Re: Geometric Sequences and Series

Hey JC, 

Good question.  Let's just look at the sequence we have before us, 
without looking at the usual methods and formulas for Geometric 
Sequence and Series analysis:

     -3072, 1536, -768

I don't know if it's straightaway obvious, but each term seems to be 
half of the previous term. To be more accurate, it seems each term is 
the negative of half of the previous term. In terms of math-speak if 
we have an arbitrary term X_n then:

          X_(n+1)    = (-1/2) * X _n

The subscript n is necessary to dictate which element follows which. 
Therefore X with a subscript 1 would be the first element in the 
sequence. So with some basic notation and with a general gist of 
what's happening let's see if we can obtain a general formula for any 
given term in the sequence. Now we know that each term is multiplied 
by -1/2 to obtain the subsequent term. Normally we call the number in 
a geometric sequence that each term is multiplied by r, which is 
really short for the term 'the common ratio'. So we know the common 
ratio, r = -1/2. We also know another important term, the first term, 
a = -3072. From the first term and the common ratio we can find the 
second term, it's simply the first term times r. Therefore 

     X_2  = ar = -3072 * (-1/2) = 1536. 

In a similar way we find that the third term = 1536 * (-1/2) = -768 
but also

     X_3 = ar * r = ar^2

and also the fourth term would be

     X_4 = ar^2 * r = ar^3

The trend is apparent if X_3 = ar^2 and X_4 = ar^3 then X_13  = ar^12.
                          3            4             13 

So the formula for X_n  = ?  [I leave this to you.]

This should enable you to answer part (a) of your assignment. But part 
(b) seems either really long or just more complex. Let me explain...

The really long and tedious way to find the sum of the first nine 
terms would be to find each term and do the addition:

     X_1 + X_2 + X_3 + X_4 + X_5 + X_6 + X_7 + X_8 + X_9 = ?

     = a + ar + ar^2 + ar^3 + ar^4 + ar^5 + ar^6 + ar^7 + ar^8 = ?

Now that would be tedious, especially if you had to do this on a test 
or something. Luckily there's a shorter though a little more complex 
way to do it. There is a formula for a geometric series sum (the 
derivation of this formula I won't go into):

     S(n) = a(r^n -1)/(r-1) if r > 1
or
     S(n) = a(1 - r^n)/(1-r) if r < 1

where n is the number of terms you want to go up to. For instance, in 
the case of this question n would be 9 since we wish to find the sum 
of the first 9 terms. I hope all this helped. Write back if anything 
is unclear. 

- Doctor Dwayne, The Math Forum
  http://mathforum.org/dr.math/