Complete a repeating pattern

Learning Objectives Identify the core repeating unit and determine the period of a complex numerical or algebraic pattern. Apply modular arithmetic to accurately predict the nth term of a repeating sequence. Solve problems involving the cyclicity of units digits in powers of integers. Analyze and predict the outcome of a sequence of repeating geometric transformations. Formulate a general expression to find any term in a repeating pattern. Distinguish between arithmetic/geometric sequences and repeating (periodic) sequences. If a traffic light cycles Green-Yellow-Red every 90 seconds, what color will it be in 400 seconds? 🚦 Repeating patterns are the hidden logic behind everything from music to computer code! This tutorial moves beyond simple shape patterns to explore the...

TermDefinitionExample SequenceAn ordered list of numbers, shapes, or other elements. In this context, we focus on sequences where elements repeat in a predictable cycle.The sequence of days of the week: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday, Monday... Pattern CoreThe smallest, non-repeating block of elements that forms the basis of the entire repeating pattern.In the pattern A, B, C, A, B, C, A, B, C..., the pattern core is (A, B, C). Period (or Cycle Length)The number of terms in the pattern core. It is the length of one full cycle before the pattern begins to repeat.For the sequence 5, 0, 5, 0, 5, 0..., the pattern core is (5, 0) and the period is 2. Term (a_n)A specific element in a sequence. The notation a_n refers to the term at the nth position.In the sequen...

Finding the Position within the Core k = n \pmod{P} To find which term in the core corresponds to the nth term of the sequence, calculate the remainder 'k' when the term number 'n' is divided by the period 'P'. This tells you the position within the cycle. Be cautious with remainders of 0. The Nth Term Rule a_n = \begin{cases} a_k & \text{if } n \pmod{P} = k \neq 0 \\ a_P & \text{if } n \pmod{P} = 0 \end{cases} The nth term of the sequence (a_n) is equal to the kth term of the pattern core, where k is the remainder of n divided by the period P. If the remainder is 0, it means the term is the last one in a complete cycle, so you use the Pth term of the core.