Evaluate explicit formulas for sequences

Learning Objectives Define a sequence and an explicit formula. Differentiate between the term number (n) and the term value (a_n). Calculate the first 'k' terms of a sequence given its explicit formula. Find the value of any specific term (e.g., the 20th term) in a sequence using its explicit formula. Correctly apply the order of operations when substituting values into linear, quadratic, exponential, and alternating explicit formulas. Generate a set of ordered pairs (n, a_n) from an explicit formula to prepare for graphing a sequence. Ever wonder how your savings account grows or how video game animations are timed? It all starts with a predictable pattern! 🔢 This tutorial will teach you how to use 'explicit formulas'—powerful recipes for finding any t...

TermDefinitionExample SequenceAn ordered list of numbers, often following a specific pattern or rule.The list 3, 7, 11, 15, ... is a sequence where each number is 4 more than the previous one. TermEach individual number in a sequence.In the sequence 3, 7, 11, 15, ..., the number 11 is the third term. Term Number (n)The position of a term in a sequence. It is always a positive integer (1, 2, 3, ...).For the term 11 in the sequence 3, 7, 11, 15, ..., the term number is n = 3. Term Value (a_n)The actual value of the term at position 'n'. The subscript 'n' links the value to its position.For the sequence 3, 7, 11, 15, ..., when n=3, the term value is a_3 = 11. Explicit FormulaA rule that calculates the value of any term (a_n) in a sequence directly from its term number (n)...

The Substitution Principle To find the k-th term, substitute n = k into the formula for a_n. This is the fundamental process for evaluating any explicit formula. Identify the desired term number (e.g., 5th term means n=5), replace every 'n' in the formula with that number, and then simplify. Order of Operations (PEMDAS/BODMAS) 1. Parentheses/Brackets, 2. Exponents/Orders, 3. Multiplication/Division (left to right), 4. Addition/Subtraction (left to right). When a formula has multiple operations, you must follow the correct order to get the right answer. This is especially critical in formulas with exponents, negatives, and fractions.