A finite sequence such as
 

in which each term after the first is obtained by multiplying the preceding term by a fixed number, is called a geometric progression. The general form of a geometric progression with n terms is therefore
 

Example. Sum 

Solution: Here the ratio r is 2² = 4. We let S designate the desired sum and write S and rS as follows:

Subtracting, we note that all but two terms on the right cancel out and we obtain
 

 

or
 

Hence we have the compact expression for the sum:
 

S =

If the ratio r is negative, the terms of the geometric progression alternate in signs. Such a progression with a = 125, r = -1/5, and n = 8 is
 

The geometric mean of two positive real numbers a and b isthe positive square root of their product; the geometric mean of three positive numbers a, b, and c is In general, the geometric mean of n positive numbers is the nth root of their product. For example, the geometric mean of 2, 3, and 4 is 
 

Sunday, March 15, 1998