![]() 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. Suppose John tossed a die 100 times and recorded the outcomes. The terms of a sequence don”t have to follow any regular pattern or rule. The numbers in the list are called the terms of the sequence. A sequence can be finite, such as 2, 5, 10, 17, 26, or can be infinite, such as 2, 4, 6, 8, 10. Most often, the objects are numbers, but they don”t have to be. ![]() ![]() In this section, you will read about the three different types of sequences that could be the source of a Math 1 test question.Ī sequence is a list of objects separated by commas. On the Math 1 test, there is often one and occasionally two questions about sequences. If the base is a Lucas number (≥ 4) with an odd index, the period is twice the index.SAT SUBJECT TEST MATH LEVEL 1 MISCELLANEOUS TOPICS That is, if the modulo base is a Lucas number (≥ 3) with an even index, the period is four times the index. If the base is a Fibonacci number (≥ 5) with an odd index, the period is four times the index and the cycle has four zeros.įirst half of cycle (for even k ≥ 4) or first quarter of cycle (for odd k ≥ 4) or all cycle (for k ≤ 3) That is, if the modulo base is a Fibonacci number (≥ 3) with an even index, the period is twice the index and the cycle has two zeros. The first 144 Pisano periods are shown in the following table: The first twelve Pisano periods (sequence A001175 in the OEIS) and their cycles (with spaces before the zeros for readability) are (using hexadecimal cyphers A and B for ten and eleven, respectively): Π( n) ≤ 6 n, with equality if and only if n = 2 The multiplicative property of Pisano periods imply thus that It follows from above results, that if n = p k is an odd prime power such that π( n) > n, then π( n)/4 is an integer that is not greater than n. That is r 2( p+1) = 1, and the Pisano period, which is the order of r, is the quotient of 2( p+1) by an odd divisor. (sequence A000045 in the OEIS)ĭefined by the recurrence relation F 0 = 0 exchanges these roots, it follows that, denoting them by r and s, we have r p = s, and thus r p+1 = –1. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. ![]() In number theory, the nth Pisano period, written as π( n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats.
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