Overview
This lecture covers arithmetic and geometric sequences, means, formulas for terms and sums, differences between sequences and series, and includes practice problems on identifying, writing, and summing various sequences.
Arithmetic vs. Geometric Sequences
- Arithmetic sequence: each term increases/decreases by a fixed amount (common difference).
- Geometric sequence: each term multiplies/divides by a fixed number (common ratio).
- Arithmetic example: 3, 7, 11, 15, 19 (common difference: 4).
- Geometric example: 3, 6, 12, 24, 48 (common ratio: 2).
Means in Sequences
- Arithmetic mean: average of two numbers, (a + b) / 2, gives the middle term in an arithmetic sequence.
- Geometric mean: square root of product of two numbers, √(a × b), finds middle term in a geometric sequence.
General Formulas
- nth term of arithmetic sequence: aₙ = a₁ + (n−1)d.
- nth term of geometric sequence: aₙ = a₁ × rⁿ⁻¹.
Partial Sums
- Arithmetic sequence sum: Sₙ = [(a₁ + aₙ) / 2] × n.
- Geometric sequence sum: Sₙ = a₁ × [1 − rⁿ] / (1 − r).
Sequences vs. Series
- Sequence: list of numbers (finite or infinite).
- Series: sum of a sequence (finite or infinite).
- Use ellipses (...) to indicate infinity.
Identifying Types and Properties
- Find common difference (arithmetic) by subtraction or common ratio (geometric) by division.
- Determine if sequence/series is finite or infinite by presence of an end or ellipses.
Practice Problems
- Write first n terms: plug values into sequence formula or use given rules.
- Find next terms: add common difference or multiply by common ratio.
- Write explicit formulas: use first term and difference/ratio, distribute and combine terms as needed.
- Calculate specific term: substitute n into explicit formula.
- Calculate sum: use appropriate sum formula after finding n.
Example Calculations
- Sum of first 300 natural numbers: S = [(1 + 300)/2] × 300 = 45,150.
- Sum of all even numbers from 2 to 100: S = [(2 + 100)/2] × 50 = 2,550.
- Find n for a term: set aₙ = a₁ + (n−1)d, solve for n.
Key Terms & Definitions
- Arithmetic Sequence — sequence with a constant difference between terms.
- Geometric Sequence — sequence with a constant ratio between terms.
- Arithmetic Mean — (a + b) / 2, average of two numbers.
- Geometric Mean — √(a × b), mean in multiplicative sequences.
- Explicit Formula — formula to find any term directly.
- Recursive Formula — formula using previous term(s) to find next.
- Series — sum of sequence terms.
- Finite/Infinite — has an end/continues forever.
Action Items / Next Steps
- Write formulas and practice finding terms and sums for given sequences.
- Review definitions and formulas for sequences and series.
- Complete assigned practice problems on arithmetic and geometric sequences.