﻿ ﻿REPEATING SEQUENCES - Sequences - MISCELLANEOUS TOPICS - SAT SUBJECT TEST MATH LEVEL 1

## MISCELLANEOUS TOPICS ## CHAPTER 18Sequences • Repeating Sequences

• Arithmetic Sequences

• Geometric Sequences

• Exercises

On the Math 1 test, there is often one and occasionally two questions about sequences. In this section, you will read about the three different types of sequences that could be the source of a Math 1 test question.

A sequence is a list of objects separated by commas. Most often, the objects are numbers, but they don”t have to be. A sequence can be finite, such as 2, 5, 10, 17, 26, or can be infinite, such as 2, 4, 6, 8, 10, . . . The numbers in the list are called the terms of the sequence.

The terms of a sequence don”t have to follow any regular pattern or rule. Suppose John tossed a die 100 times and recorded the outcomes. If you saw the first 10 terms of that sequence—4, 6, 6, 2, 5, 3, 5, 4, 4, 2—you would have no way of knowing what the 11th term is.

On the Math 1 test, however, there is always a definite rule that determines the terms of a sequence. If Mary created a sequence of 100 terms by writing down the number 5 and then continually added 3 to each term, the first 10 terms would be 5, 8, 11, 14, 17, 20, 23, 26, 29, 32. Not only would you be able to know what the eleventh term is, as you will see, you can easily calculate the 88th, or any other, term.

On the Math 1 test, the most common type of question concerning sequences gives you a rule and asks you to find a particular term. The three types of sequences that occur most often are repeating sequences, arithmetic sequences, and geometric sequences.

### REPEATING SEQUENCES

A sequence whose terms repeat in a cyclical pattern is called a repeating sequence. For example, the following three sequences are repeating sequences:

a, b, c, a, b, c, a, b, c, a, b, c, . . .

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, . . .

7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, . . .

Key Fact Q1

When a sequence consists of a group of k terms that repeat in the same order indefinitely, to find the nth term, find the remainder, r, when n is divided by k. The rth term and the nth term are equal.

The fraction is equivalent to the repeating decimal 0.714285714285 . . . in which the six digits, 7, 1, 4, 2, 8, 5, repeat indefinitely. Examples 1 and 2 refer to this sequence of digits.

EXAMPLE 1: To find the 1,000th digit to the right of the decimal point in the expansion of , use KEY FACT Q1. Since the repeating portion consists of 6 digits, divide 1,000 by 6. 1,000 ÷ 6 = 166.666 . . . the quotient is 166
166 × 6 = 996 the remainder is 1,000 – 996 = 4

Therefore, the 1,000th term is the same as the 4th term, namely 2.

EXAMPLE 2: What is the sum of the 101st through the 106th digits in the decimal expression of ?

You could repeat what was done in Example 1 six times, but, of course, you shouldn”t. Any six consecutive terms of this sequence consist, in some order, of the same six digits—7, 1, 4, 2, 8, 5—whose sum is 27. In this case, the order is 8, 5, 7, 1, 4, 2, but you do not need to know that.

At some point in your study of math you may have seen the sequence 1, 1, 2, 3, 5, 8, 13, . . . This sequence, called the Fibonacci sequence, is defined by the following rule: the first two terms are both 1 and, starting with the third term, each term is the sum of the two preceding terms. For example, 2 = 1 + 1; 13 = 5 + 8; and since the sum of the 6th and 7th terms, 8 and 13, is 21, the 8th term is 21. Clearly this is not a repeating sequence, and it would be totally unreasonable to ask you to find the 100th term. However, there are some questions, such as the one in Example 3, that you could be asked.

EXAMPLE 3: Of the first 100 terms of the Fibonacci sequence, how many are odd? The sequence itself does not form a repeating sequence, but its pattern of odd (O) and even (E) terms does:

 1 1 2 3 5 8 13 21 34 O O E O O E O O E

Note that the Os and Es form a repeating sequence: The first 99 terms consist of 33 sets of . Since each set contains two Os and one E, of the first 99 terms, 66 are odd and 33 are even. The 100th term is the first term in the next set and so is O. In all, there are 67 odd terms.

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