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### Using the Geometric Mean in a Sequence

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Date: 07/11/2000 at 02:51:58
From: Kari
Subject: Geometric Sequence and Square Roots

Discrete Mathematics:

In the geometric sequence: 5,15,_,135,405, the missing number is
called the geometric mean of 15 and 135. It can be found by
evaluating the square root of ab, where a and b are the numbers on
either side of the geometric mean. Find the missing number.

I do not know what the geometric mean is, so I'm guessing it is the
middle factor in the list of factors for a number. If so, how can it
help solve this problem?

Also, how can you find the square root of ab if you do not know what
a and b represent? How can that help solve the equation?

Also, I know this is sort of off the problem, but how do you find the
square root of a number without using a calculator?

Thank you.
```

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Date: 07/11/2000 at 11:29:12
From: Doctor Rick
Subject: Re: Geometric Sequence and Square Roots

Hi, Kari, thanks for writing.

The problem is trying to tell you exactly what to do, but it's not
getting its point across to you, so let's say it another way.

You know what a geometric sequence is, right? Each number is some
constant times the number to its left. In the sequence that you are
given, the second number (15) is 3 times the first number (5). Since
it is a geometric sequence, you know that each number in the sequence
will follow the same rule. This means that the third number (which is
missing) is 3 times the second number (15), so it is 45. The fourth
number is 3 times 45, which is 135, and so on.

I've already solved the problem for you - I found the missing number,
45. But the problem is teaching you another way to find the missing
number, without the need to find the constant factor in the sequence.

The problem says that in a geometric series, each number is the
geometric mean of the numbers on either side of it. This is the
DEFINITION of the geometric mean: it's the number that goes between
two other numbers in a geometric sequence. Then it tells how to
calculate the geometric mean of two numbers: it is the square root of
the product of the two numbers. For instance, the geometric mean of 4
and 9 is the square root of 4*9. Since 4*9 = 36 and the square root of
36 is 6, the geometric mean of 4 and 9 is 6.

The problem says that the geometric mean can be found by evaluating
the square root of ab, where a and b are the numbers on either side of
the geometric mean. That's just another way of saying what I said in
the last paragraph. Now, what you want to do is to find the number in
the blank, which is the geometric mean of the numbers on either side
of it. What are those numbers? They are 15 and 135. You DO know what a
and b are: a is 15 and b is 135, so you can work out the geometric
mean.

I'll give you a little extra; you don't need this to solve the
problem, but I hope you're curious about it. WHY is the geometric mean
calculated this way? Let's do a little algebra.

Let's use the variable p to represent one number in a geometric
sequence, and r to represent the constant factor in the sequence. Then
the number after p in the sequence is p*r (multiply p by the factor r
to get the number after p). The number after that is p*r*r, or p times
r squared. Now we have 3 numbers in a row in the series:

p, p*r, p*r*r

The geometric mean of the two outside numbers, p and p*r*r, is

sqrt(p * p*r*r)

= sqrt(p*p)*sqrt(r*r)

= p*r

which is the middle number. So you see that the middle number is the
geometric mean of the two numbers on either side of it.

calculate square roots without a calculator in our Dr. Math FAQ. Look
at

Square/cube roots without a calculator
http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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