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 57 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.

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.