Finding the Next Number in a Sequence Given Its Geometric Mean ... Which Is a Square Root

Date: 09/24/2009 at 18:45:27
From: Brittany
Subject: geometric mean, finding 2nd number when you have the sq root

What is the next number in a sequence if the first number is 12 and 
you are given the sequence's geometric mean?

I have this same problem with the first number being 4.

I understand how to find the geometric mean of a pair of numbers; but 
starting with SQRT(6) has me all confused.  I do not understand how 
to find the second number.

Every calculation I have tried comes up with an even number, not a 
decimal form.  I have tried 6*24, I have tried adding the numbers, ...

I am going through an online program and they are not good at 
explaining the correct way to figure this.

Date: 09/25/2009 at 09:39:38
From: Doctor Ian
Subject: Re: geometric mean, finding 2nd number when you have the sq 
root

Hi Brittany,

When in doubt, go back to definitions.

In an arithmetic sequence, we get each new term by ADDING a constant 
amount to the preceding term.  For example,

   5, 8, 11, 14, 17, ...

is a sequence that we get by starting with 5, and adding 3 to each 
term to get the next term.

The arithmetic mean is the number 'in the middle' of two other 
numbers in an arithmetic sequence.  To get it, we add the numbers and 
divide by 2:

  5 + 11        8 + 14         11 + 17
  ------ = 8    ------ = 11    ------- = 14
    2             2               2

and so on.  This amounts to the same thing as 'splitting the 
difference,' e.g., 

       11 - 5
   5 + ------ = 5 + 3 = 8
          2

because the distance between the two numbers will be twice the 
constant term.

Does this make sense?  Let me know if it doesn't. 

In a geometric sequence, we get each new term by MULTIPLYING a 
constant amount by the preceding term.  For example, 

   5, 15, 45, 135, 405, ...

is a sequence that we get by starting with 5, and multiplying 3 by 
each term to get the next term.
 
The geometric mean is the number 'in the middle' of two other numbers 
in an geometric sequence.  (You can sort of see where this is going, 
right?)

So, how can we get the geometric mean?  Well, suppose we're looking 
at a part of a geometric sequence,

   a, ?, b

We know that there is some value k such that

   ? = k*a             ?/a = k

   b = k*?    -->      b/? = k

That follows from the definition of a geometric sequence.  So we have 
two expressions for the quantity k, which must be equal to each other:

   ?/a = b/?

We can cross multiply, to get

   ?^2 = ab
           ___
     ? = \/ ab

In our sequence, 

   5, 15, 45, 135, 405, ...
 
we get
     _____           _______            _______
   \/ 5*45 = 15    \/ 15*135 = 45     \/ 45*405 = 135

and so on.

So to find the geometric mean of two quantities, we take the square 
root of their product.  It's not nearly as intuitive as the 
arithmetic mean, but do you see how it follows from the same kind of 
reasoning?

Another way to see this is to write the terms as

   a, k*a, k^2*a

That tells us that 
   
       b = k^2 * a

     b/a = k^2
    ____
  \/ b/a = k

This gives us the constant directly, and is a little more like the 
way we 'split the difference' with the arithmetic sequence:

  * In an ARITHMETIC sequence, 
    we ADD a constant twice, 
    so we HALVE the resulting change 
    (i.e., the DIFFERENCE of our terms) to find the constant.   

  * In a GEOMETRIC sequence,
    we MULTIPLY by a constant twice, 
    so we take the SQUARE ROOT of the resulting change 
    (i.e., the RATIO of our terms).  

So it's really the same reasoning, just using different operations. 
The nice thing about seeing that is, if you forget the particular 
rules, you can re-create them by starting from the basic ideas -- 
namely, that you add or multiply, and you want to find the term in 
the middle of two other terms.

Does this help? 

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