Overview
This lecture explains the differences between sequences and series, explores arithmetic and geometric sequences, their formulas (explicit and recursive), methods to find sums—including finite and infinite cases—and introduces summation notation with example problems.
Sequences vs. Series
- A sequence is a list of numbers separated by commas; a series is the sum of a sequence's terms.
- Each term in a sequence can be identified by its position (n) and written as aₙ.
Arithmetic Sequences
- Arithmetic sequences add the same value (common difference, d) between terms.
- The nth term: aₙ = a₁ + (n - 1)d.
- For recursive formulas: a₁ is given, then aₙ = aₙ₋₁ + d.
- To find the sum of the first n terms: Sₙ = n/2 × (a₁ + aₙ).
Geometric Sequences
- Geometric sequences multiply by the same value (common ratio, r) between terms.
- The nth term: aₙ = a₁ × rⁿ⁻¹.
- Recursive formula: a₁ given, then aₙ = aₙ₋₁ × r.
- Sum of first n terms: Sₙ = a₁ × (1 - rⁿ) / (1 - r), r ≠ 1.
Infinite Series
- For a geometric series where |r| < 1, the sum to infinity: S∞ = a₁ / (1 - r).
- If |r| ≥ 1, the series diverges and no finite sum exists.
Solving for Sequence Rules Given Terms
- For arithmetic sequences, use given terms to set up equations and solve for a₁ and d.
- For geometric sequences, use given terms to set up equations, solve for a₁ and r.
Summation Notation (Sigma)
- Σ (sigma) represents the sum of terms from the lower index to the upper index.
- Plug each index value into the formula and add results sequentially.
Counting Terms in Summation
- The number of terms from index a to b is (b - a) + 1.
Example Applications
- Use explicit or recursive formulas to find specific terms or the sum for arithmetic/geometric sequences.
- For summation notation, carefully substitute index values and use appropriate formula to sum.
Key Terms & Definitions
- Sequence — Ordered list of numbers.
- Series — Sum of terms in a sequence.
- Arithmetic Sequence — Sequence with a constant difference (d) between terms.
- Geometric Sequence — Sequence with a constant ratio (r) between terms.
- Explicit Formula — Directly calculates the nth term.
- Recursive Formula — Defines each term based on a previous term.
- Common Difference (d) — The value added in an arithmetic sequence.
- Common Ratio (r) — The value multiplied in a geometric sequence.
- Summation Notation (Σ) — Notation for summing a sequence of terms.
- Convergent Series — Infinite series that approaches a finite sum.
- Divergent Series — Infinite series that increases or decreases without bound.
Action Items / Next Steps
- Practice writing explicit and recursive formulas for given sequences.
- Solve example problems using summation notation.
- Review formulas for finite and infinite arithmetic and geometric series.