Evaluate variable expressions for number sequences

Learning Objectives Identify patterns in number sequences. Write variable expressions to represent the nth term of a sequence. Substitute given term positions into variable expressions. Accurately evaluate variable expressions to find specific terms in a sequence. Apply the order of operations when evaluating expressions for sequences. Distinguish between the term number (position) and the term value. Ever wondered how scientists predict the population of a bacteria colony or how much money you'll save each month? 📈 It often involves understanding patterns and using math to make predictions! In this lesson, you'll learn how to describe these patterns using variable expressions and then use those expressions to find any term in a sequence. This skill is fundamenta...

TermDefinitionExample Number SequenceAn ordered list of numbers that follows a specific pattern or rule.The sequence 2, 4, 6, 8, ... is an ordered list of even numbers. TermEach individual number in a sequence.In the sequence 2, 4, 6, 8, ..., '2' is the first term, '4' is the second term, and so on. Position (Term Number)The place or order of a term within a sequence, typically denoted by the variable 'n'.In the sequence 2, 4, 6, 8, ..., the term '6' is at position n=3. Variable ExpressionA mathematical phrase that contains variables (like 'n'), numbers, and at least one operation, used to represent the rule of a sequence.The expression '2n' can represent the nth term of the sequence of even numbers. Evaluating an ExpressionThe p...

Rule for Evaluating Variable Expressions 1. Substitute the given numerical value for the variable (usually 'n' for term position). 2. Simplify the expression using the order of operations (PEMDAS/BODMAS). This rule is used whenever you have a variable expression for a sequence and need to find the value of a specific term (e.g., the 5th term, the 100th term) by plugging in its position. General Form for an Arithmetic Sequence (Linear Expression) For an arithmetic sequence, the $n$-th term, $a_n$, can often be represented by a linear expression of the form $a_n = dn + c$, where $d$ is the common difference and $c$ is a constant. This rule helps you write a general expression for any arithmetic sequence. Once you find the common difference ($d$) and the constant ($c$...