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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, . . . ?

  1. 1/3
  2. 0
  3. –10/27
  4. –3
  5. –100/3
Solution

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

 

Hence, the sixth number in the sequence is .

 

The answer is (C).
 

Advanced concepts: (Sequence Formulas)

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

 

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




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