# Geometric Progressions

A geometric progression is a sequence in which the ratio of any two consecutive terms is the same. Thus, each term is generated by multiplying the preceding term by a fixed number.
For example, –3, 6, –12, 24, . . . is a geometric progression in which the common ratio is –2. The sequence 32, 16, 8, 4, . . . is geometric with common ratio 1/2.

Example

What is the sixth term of the sequence 90, –30, 10, –10/3, . . . ?
A.  1/3
B.  0
C.  –10/27
D.  –3
E.  –100/3

Since the common ratio between any two consecutive terms is –1/3, the fifth term is .

Hence, the sixth number in the sequence is .

Note: none of the formulas in this section are necessary to answer questions about sequences on the test.

Since each term of a geometric progression “is generated by multiplying the preceding term by a fixed number,” we get the following:

 first term a second term where r is the common ratio third term fourth term . . . nth term This formula generates the nth term

The sum of the first n terms of an geometric sequence is