Evaluate variable expressions for number sequences

Learning Objectives Define a number sequence and the variable expression that generates it. Substitute a given term number (n) into a variable expression to find the value of that term. Evaluate linear expressions (e.g., an + b) for given values of n. Evaluate quadratic expressions (e.g., an^2 + bn + c) for given values of n. Evaluate expressions involving radicals or exponents for given values of n. Generate the first 'k' terms of a sequence given its variable expression. Ever wonder how video games create progressively harder levels or how your savings account grows over time? 📈 It's all about predictable patterns called sequences! This tutorial will teach you how to use a formula, called a variable expression, to find any specific number in a pattern or s...

TermDefinitionExample Number SequenceAn ordered list of numbers, where each number follows a specific rule or pattern.The list 5, 8, 11, 14, ... is a number sequence where each number is 3 more than the previous one. TermEach individual number in a sequence. We often use a subscript to denote the term, like T_n.In the sequence 5, 8, 11, 14, ..., the third term (T_3) is 11. Term Number (n)The position of a term in a sequence. It's almost always a positive integer starting from n=1.For the term 11 in the sequence 5, 8, 11, 14, ..., the term number is n=3. Variable Expression (Rule)An algebraic formula that describes the relationship between a term's value and its position (n) in the sequence.The expression 3n + 2 generates the sequence 5, 8, 11, 14, ... EvaluateTo substitute a spe...

Substitution Principle To evaluate an expression for a variable, replace every instance of the variable with its given numerical value. This is the fundamental action for this topic. If you are asked to find the 10th term (n=10) of a sequence with the rule T_n = 5n - 1, you replace 'n' with '10' to get T_10 = 5(10) - 1. General Form of a Linear Expression T_n = an + b This rule generates arithmetic sequences where there is a constant difference between terms. To find the value of any term (T_n), you substitute its position (n) into the formula. General Form of a Quadratic Expression T_n = an^2 + bn + c This rule generates sequences where the second difference is constant. To evaluate, you must follow the order of operations, calculating the expone...