Overview
This lecture covers arithmetic and geometric sequences and series, including their definitions, formulas, how to distinguish between them, and how to calculate terms and sums.
Arithmetic vs. Geometric Sequences
- Arithmetic sequence: each term increases/decreases by a constant difference.
- Geometric sequence: each term is multiplied/divided by a constant ratio.
- Example arithmetic: 3, 7, 11, 15, 19 (common difference = 4).
- Example geometric: 3, 6, 12, 24, 48 (common ratio = 2).
Means in Sequences
- Arithmetic mean: (a + b)/2, gives the middle term in arithmetic sequences.
- Geometric mean: √(ab), gives the middle term in geometric sequences.
Finding Terms in Sequences
- Arithmetic nth term: aₙ = a₁ + (n – 1)d.
- Geometric nth term: aₙ = a₁ × rⁿ⁻¹.
- Example: Find 5th term of arithmetic sequence with a₁ = 3, d = 4: a₅ = 3 + (5–1)×4 = 19.
- Example: Find 6th term of geometric sequence with a₁ = 3, r = 2: a₆ = 3 × 2⁵ = 96.
Sums in Sequences (Series)
- Arithmetic series sum: Sₙ = (a₁ + aₙ)/2 × n.
- Geometric series sum: Sₙ = a₁ × (1 – rⁿ) / (1 – r).
- Example: Sum of first 7 terms (arithmetic) with a₁ = 3, a₇ = 27: S₇ = (3+27)/2×7 = 105.
Sequence vs. Series & Finite vs. Infinite
- Sequence: ordered list of numbers.
- Series: sum of numbers in a sequence.
- Finite: has a definite end; Infinite: continues without end.
Identifying and Analyzing Sequences
- Arithmetic: common difference, e.g., 4, 7, 10, 13.
- Geometric: common ratio, e.g., 4, 8, 16, 32.
- Neither: no constant difference or ratio.
Recursive and Explicit Formulas
- Recursive: aₙ = aₙ₋₁ + d or specific function using previous term.
- Explicit: direct formula for nth term using n.
Writing Sequence Formulas
- Identify a₁ and d (arithmetic) or r (geometric).
- Write explicit formula: aₙ = a₁ + (n-1)d (arithmetic).
- For fractions, write numerator and denominator formulas separately.
Practice Problem Approaches
- To write terms: plug n into the formula or apply the difference/ratio.
- For sums: use the appropriate Sₙ formula.
- To find n given last term: solve aₙ = a₁ + (n–1)d for n.
Key Terms & Definitions
- Arithmetic sequence — sequence with constant difference between terms.
- Geometric sequence — sequence with constant ratio between terms.
- Arithmetic mean — average of two numbers: (a + b)/2.
- Geometric mean — square root of their product: √(ab).
- Explicit formula — formula to find any term directly.
- Recursive formula — formula using previous term(s) to find the next.
- Series — sum of sequence terms.
- Finite/Infinite — whether a sequence or series ends or continues forever.
Action Items / Next Steps
- Practice writing arithmetic and geometric sequences from a formula.
- Memorize key formulas for nth term and sums of sequences.
- Complete assigned problems on identifying, classifying, and summing sequences/series.