Overview
This lecture covers the differences between sequences and series, how to identify and work with arithmetic and geometric sequences, and how to find sums using various formulas, including finite and infinite series.
Sequences vs. Series
- A sequence is an ordered list of numbers, separated by commas.
- A series is the sum of the terms in a sequence.
Notation and Terms
- Terms are denoted as a₁ (first term), a₂ (second term), etc.; subscript shows position.
- n represents the term number; for example, n=6 is the sixth term.
Arithmetic Sequences
- Arithmetic sequences add a constant value (common difference, d) each time.
- Common difference: d = (any term) - (previous term).
- General formula: aₙ = a₁ + d(n-1).
- To find the nth term, add d (n-1) times to the first term.
- Recursive formula: a₁ = first term; aₙ = aₙ₋₁ + d.
Arithmetic Series
- The sum of the first n terms: Sₙ = n/2 × (a₁ + aₙ).
- Used when you know the first and last terms.
- For the sum of numbers 1 to 100: S = 100/2 × (1+100) = 5050.
Geometric Sequences
- Geometric sequences multiply by a constant ratio (r) each time.
- Common ratio: r = (any term) ÷ (previous term).
- General formula: aₙ = a₁ × rⁿ⁻¹.
- Recursive formula: a₁ = first term; aₙ = aₙ₋₁ × r.
Geometric Series
- Sum of first n terms: Sₙ = a₁ × [1 - rⁿ] / (1 - r), for |r| ≠ 1.
- For infinite series (|r| < 1): S = a₁ / (1 - r).
Special Cases & Problem Solving
- To find an explicit formula given two terms, use equations to solve for a₁ and d (arithmetic) or a₁ and r (geometric).
- Summation notation (Σ) is used to represent adding a sequence’s terms.
- When counting terms from x to y inclusively: total terms = y - x + 1.
Key Terms & Definitions
- Sequence — An ordered list of numbers.
- Series — The sum of a sequence’s terms.
- Arithmetic Sequence — Sequence where each term increases by a constant difference.
- Geometric Sequence — Sequence where each term is multiplied by a constant ratio.
- Common Difference (d) — The fixed amount added in an arithmetic sequence.
- Common Ratio (r) — The fixed number multiplied in a geometric sequence.
- Recursive Formula — Defines each term based on previous term(s).
- Explicit Formula — Formula to find any term directly by its position.
Action Items / Next Steps
- Practice identifying and writing explicit and recursive formulas for given sequences.
- Complete homework problems using arithmetic and geometric series formulas.
- Review summation notation and practice evaluating sums.