Overview
This lecture introduces geometric sequences, explains how to find the common ratio, determine if a sequence is geometric, calculate terms, and solve related problems.
Geometric Sequences and Common Ratio
- A geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio (r).
- The common ratio (r) is found by dividing any term by the preceding term.
- If all consecutive terms share the same ratio, the sequence is geometric.
Identifying and Extending Geometric Sequences
- To find the next term, multiply the last term by the common ratio.
- Example: For the sequence 1, 2, 4, 8, the common ratio is 2, so the next term is 16 (8 × 2).
- Sequences not sharing a constant ratio are not geometric.
Checking for Geometric Sequences
- Test for a geometric sequence by checking if the ratio between all consecutive terms is constant.
- Example: 5, 20, 80, 320 has a common ratio of 4 (geometric).
- Example: 5, -10, 20, -40 has a common ratio of -2 (geometric).
- Sequences with non-constant ratios are not geometric.
Formula for the nth Term
- The nth term formula: aₙ = a₁ × r^(n-1).
- a₁ is the first term, r is the common ratio, n is the term number, and r ≠ 0.
- Example: To find the 10th term of 8, 4, 2, 1, use 8 × (1/2)^(10-1) = 1/64.
Solving Geometric Problems
- To find a missing term, multiply the previous term by the common ratio.
- For real-life applications, like disease outbreaks, use the formula to predict future terms.
Key Terms & Definitions
- Geometric Sequence — a sequence with a constant ratio between consecutive terms.
- Common Ratio (r) — the fixed number each term is multiplied by to get the next term.
- nth Term (aₙ) — the value at position n in the sequence, calculated by aₙ = a₁ × r^(n-1).
Action Items / Next Steps
- Practice identifying geometric sequences and finding the common ratio.
- Use the formula aₙ = a₁ × r^(n-1) to solve for requested terms.
- Complete assigned exercises on finding missing terms in geometric sequences.