**Arithmetic, Geometric, and Harmonic Means**

**By: Donna Kharrazi**

History

In 500 B.C. Pythagoras and other Greek mathematicians studied Arithmetic, Geometric, and Harmonic means.

They were developed with references to music theory, geometry, and arithmetic.

They are the three classical Pythagorean means.

Arithmetic Means

Terms between any two nonconsecutive terms of an arithmetic sequence.

It is also the average between two numbers

Each number in an arithmetic sequence is the average of the numbers on its side. Ex/10,2,-6... (10+(-6))/ 2 = 2

Geometric Means

Terms between any two nonconsecutive terms of a geometric sequence.

The average is the "nth" root of the product of "n" numbers.

To find the average use...

**Harmonic Means**

Terms between any two nonconsecutive numbers of a harmonic sequence.

The reciprocal of arithmetic means.

If in the semicircle ADC, with center O, has DB perpendicular to AC and BF perpendicular to DO, then DO is the arithmetic mean, DB the geometric mean, and DF the harmonic mean (Journal of Statistics Education).

To find the average, use...

or

To find the terms between two numbers... use the explicit formula to solve for "d". Then, plug it back in to the equation to get the desired terms.

or

To find the terms between two nonconsecutive numbers... use the explicit formula of a geometric sequence to find "r". Then, plug it back into the formula to find the desired terms.

To find the average, use...

or

To calculate the terms between the two nonconsecutive numbers in a harmonic sequence...

1)Find out if it's reciprocals make an arithmetic sequence.

2)If yes, solve for the arithmetic means.

3)Find the reciprocals of those means.

**Sample Problems**

1) Find the arithmetic mean of the following numbers: 9, 3, 7, 8.

2) What is the geometric mean of 2, 8, and 4?

3) What is the harmonic mean of 2, 1/2, and 4?

#1

=

9+3+7+8

4

=

27

4

=

6.75

Add up all the numbers, then divide by 4 because there are four terms.

#2

=

=

Multiply the numbers together. Then take the cube root, because there are three terms.

#3

=

=

3

(1/2)

+ (1/1/2)

+ (1/4)

=

1.09

Add up the reciprocals of each number together. Then, divide them from 3, because there are three terms.

How does this relate to Honors Algebra 2B?

In Unit 4, we studied how to solve for arithmetic and geometric means. To find them, we used the explicit formulas of arithmetic and geometric sequences (as mentioned earlier in this Prezi). However, we did not learn about harmonic means or how to solve them. Harmonic means are the reciprocal of arithmetic means.

Real World Connection

Arithmetic means are the most commonly used/known mean out of the three.

Harmonic means are the least commonly used.

Many people use these means on a daily basis.

Arithmetic means are used in statistics, economics, sociology, and history. An example is per capita income, which is the arithmetic average of the nation's population.

Geometric means are used for proportional growth (exponential and varying growth), and social statistics. For example, if one was trying to see by what constant factor one's investment needs to be multiplied by each year in order to achieve a desired amount of money by the year 2035.

Harmonic means are used for finding the average of ratios/rates, physics, computer science, finance, trigonometry, and geometry. For example, if one if trying to find the average speed of his/her car on a 2-day road trip.

Bibliography

Algebra 2. Austin, TX: Holt, Rinehart and Winston, 2003. Print.

"Applications of the Geometric Mean." Question Corner and Discussion Area. N.p., n.d. Web. 3 May 2015.

" The Arithmetic Mean." Free Math Help. N.p., n.d. Web. 03 May 2015.

"The Early History of Average Values and Implications for Education." Journal of Statistics Education, V11N1: Bakker. Freudenthal Institute, Utrecht University, n.d. Web. 03 May 2015.

"Geometric Mean and Harmonic Mean." Geometric Mean and Harmonic Mean. John R. Schuyler, Jan. 2005. Web. 03 May 2015.

Gibilisco, Stan, and Norman H. Crowhurst. Mastering Technical Mathematics. New York: McGraw-Hill, 2008. Print.

"Harmonic Sequence, Means, and Series." Sequences and Series. N.p., n.d. Web. 03 May 2015.

"Mathwords: Arithmetic Mean." Mathwords: Arithmetic Mean. N.p., 28 July 2014. Web. 03 May 2015.

Sample Problems: Part 2

1) Find two arithmetic means between -17 and -71.

2) Find two geometric means between 5 and 135.

3) Find two harmonic means between 6 and 3/2.

#1

tn = t1 + (n-1)(d)

t4 = -17 + (4-1)(d)

-71 = -17 + 3d

d= -18

t2 = -17 + (1)(-18)

t2 = -35

t3 = -17 + (2)(-18)

t3 = -53

First, solve for "d" using the explicit formula for arithmetic sequences. Next, plug "d" back into the formula to find t2 and t3 (the two arithmetic means).

#2

tn = t1(r^n-1)

t4 = 5(r^3)

135 = 5 (r^3)

r = 3

t2 = 5(3^2-1)

t2 = 15

t3 = 5(3^3-1)

t3 = 45

First, solve for "r" using the explicit formula for geometric sequences. Next, plug "r" back into the formula and solve for t2 and t3 (the two geometric means).

#3

t4 = t1 + (n-1)(d)

2/3 = 1/6 + (3)(d)

d = 4/3

t2 = 1/6 + (1)(4/3)

t2 = 1.5

t3 = 1/6 + (2)(4/3)

t3 = 17/6

= 2/3 & 6/17

First, find the arithmetic means, by solving for "d" using the reciprocals of the numbers given. The reciprocals of those two solutions result in the two harmonic means.