Using the Geometric Mean in a Sequence

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.

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.

On your last question, you can find information about how to 
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/