I need help remembering how to work the formula to figure the 'nth' degree of a nmbr

[merl]

I need help with this question: I am studying business statistics and it has been 6 years since college algebra...
In 1996 a totoal of 14,968,000 taxpayers in the United States filed their individual tax returns electronically. By the year 2005 the number increased to 68,476,000. What is the geometric mean annual increase for the period?

The examples prior to this problem showed the formula to figure the geometric mean. What I can't remember is how to actually implement all the stuff in the formula!

 

[HallsofIvy]

The geometric mean makes sense when dealing with rates of change over multiple periods of equal length.

Example

Rate of change = + 5% EACH year

Starting position = 100.

Position at end of year 1 = 100 * 1.05 = 105.

Position at end of year 2 = 105 * 1.05 = 110.25

If I calculate the average rate of change from the starting position and ending position as: \(\displaystyle \dfrac{110.25 - 100}{100} \div 2 = 5.125\%\), which is clearly wrong.

Do it this way to get the correct answer: \(\displaystyle \left(\dfrac{110.25}{100}\right)^{1/2} - 1 = \sqrt{1.1025} - 1 = 1.05 - 1 = 5\%.\)

Why is this called a geometric mean?

Because \(\displaystyle \left(\displaystyle\prod_{i=1}^na_i\right)^{1/n} =\sqrt[n]{\displaystyle \prod_{i=1}^na_i}\ is\ the\ definition\ of\ the\ geometric\ mean\ and\)

\(\displaystyle \left(\dfrac{110.25}{100}\right)^{1/2} = \left(\dfrac{105}{100} * \dfrac{110.25}{105}\right)^{1/2}.\)

Are you good to go now?

It is called a geometric mean because if we have a right triangle with vertices A, B, C, C being the right angle (equivalently, AB is a diameter of a circle, C any other point on the circle), with legs of length a, b, and hypotenuse of length c, then dropping a perpendicular from the right angle to the hypotenuse divides the original triangle into two right triangles, both similar to the original triangle, h, the length of that perpendicular, is the "geometric mean" of a and b.