Evaluate variable expressions for number sequences

Learning Objectives Identify the variable expression (rule) for a given number sequence. Understand that the variable 'n' in an expression represents the term number or position in a sequence. Accurately substitute a given term number into a variable expression. Apply the order of operations to correctly evaluate the expression. Calculate the value of any specific term in a sequence using its variable expression. Distinguish between the term number and the value of the term in a sequence. Ever wonder how mathematicians predict what comes next in a pattern, like the number of petals on a flower or seats in an auditorium? 🌸🔢 In this lesson, you'll learn how to use special algebraic rules, called variable expressions, to find any term in a number sequence with...

TermDefinitionExample Number SequenceAn ordered list of numbers that follows a specific pattern or rule.2, 4, 6, 8, ... TermEach individual number in a sequence.In the sequence 2, 4, 6, 8, ..., '2' is the 1st term, '4' is the 2nd term. Term Number (Position)The position of a term in the sequence, usually represented by the variable 'n'.For the 3rd term, the term number 'n' would be 3. Variable Expression (Rule)An algebraic expression containing a variable (like 'n') that defines the relationship between the term number and the value of the term.For the sequence 2, 4, 6, 8, ..., the variable expression is $2n$. EvaluateTo find the numerical value of an expression by substituting a specific number for the variable and performing the indicate...

Substitution Principle $a_n = \text{Expression}(n) \implies a_k = \text{Expression}(k)$ To find the $k$-th term of a sequence, substitute the term number $k$ for the variable $n$ in the given expression. Order of Operations (PEMDAS/BODMAS) Parentheses $\rightarrow$ Exponents $\rightarrow$ Multiplication/Division $\rightarrow$ Addition/Subtraction This sequence must be followed when evaluating any mathematical expression to ensure correctness. Multiplication and Division are performed from left to right, as are Addition and Subtraction. Linear Sequence Expression Form $a_n = dn + c$ For many common sequences (arithmetic sequences), the rule that generates the $n$-th term ($a_n$) is a linear expression where $d$ is the common difference and $c$ is a constant.