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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,
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/
```
Associated Topics:
High School Definitions
High School Sequences, Series
High School Statistics
Middle School Definitions
Middle School Statistics

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