Algebra 2: Understanding Sequences

Introduction to Sequences

• A sequence is an ordered list of numbers.
• Each number in a sequence is called a term.
• The order of terms in a sequence is crucial.

Types of Sequences

Arithmetic Sequence

• Defined by a common difference between consecutive terms.
• Example: First row of chairs has 10, next rows increase by 2: 10, 12, 14, 16, etc.
• Recursive Rule: Uses the previous term to get the next term.
  • Formula: ( a_n = a_{n-1} + d )
  • Example: ( a_1 = 10, a_{n} = a_{n-1} + 2 )
• Explicit Rule: Directly calculates any term using its position.
  • Formula: ( a_n = a_1 + (n-1)d )
  • Example: ( a_n = 10 + (n-1)2 )

Geometric Sequence

• Defined by a common ratio between consecutive terms.
• Example: Start with 5, double each time: 5, 10, 20, 40, etc.
• Recursive Rule: Uses the previous term with multiplication.
• Explicit Formula: Uses position to find any term.
  • Formula: ( a_n = a_1 \times r^{n-1} )
  • Example: ( a_n = 5 \times 2^{n-1} )

Solving Sequence Problems

Arithmetic Sequence Example

• Given: First term = 80, common difference = -6.
• **Find the 10th Term:
  • Formula: ( a_n = a_1 + (n-1)d )
  • Calculation: ( a_{10} = 80 + 9(-6) = 80 - 54 = 26 )

Geometric Sequence Example

• Given: First term = 0.25, common ratio = 2.
• Find the 10th Term:
  • Formula: ( a_{10} = 0.25 \times 2^{10-1} )
  • Calculation: ( a_{10} = 0.25 \times 2^9 = 128 )

General Formulas

Arithmetic Sequence

• ( a_n = a_1 + (n-1)d )
  • ( a_1 ): First term
  • ( d ): Common difference
  • ( n ): Term number

Geometric Sequence

• ( a_n = a_1 \times r^{n-1} )
  • ( a_1 ): First term
  • ( r ): Common ratio
  • ( n ): Term number

Key Takeaways

• Arithmetic sequences involve addition/subtraction.
• Geometric sequences involve multiplication/division.
• Both recursive and explicit formulas are useful in different contexts.
• Understanding the type of sequence helps determine the formula to use.