## SAT SUBJECT TEST MATH LEVEL 1

## MISCELLANEOUS TOPICS

## CHAPTER 18

Sequences

### GEOMETRIC SEQUENCES

A ** geometric sequence** is a sequence such as 3, 6, 12, 24, 48, . . . in which the ratio between any two consecutive terms is the same. In this sequence, the ratio is

An easy way to find the *n*th term of a geometric sequence is to start with the first term and multiply it by the common ratio *n* – 1 times. Here the 5th term is 48, which can be obtained by taking the first term, 3, and multiplying it by the common ratio, 2, four times: 3 × 2 × 2 × 2 × 2 = 3 × 2^{4} = 3 × 16 = 48. In the same way, the 100th term is 3 × 2^{99}.

**Key Fact Q3**

**If a_{1}, a_{2}, a_{3}, . . . is a geometric sequence whose common ratio is r, then a_{n}** =

*a*_{1}*r*^{n-1}.**EXAMPLE 5:** What is the 12th term of the sequence 3, –6, 12, –24, 48, –96, . . . ?

This is a geometric sequence whose common ratio is –2. By KEY FACT Q3,

*a*_{12} = *a*_{1}(–2)^{11} = 3(–2)^{11} = 3(–2,048) = –6,144