Overview
This lecture explains the differences between finite and infinite geometric sequences, using clear examples and highlighting their main characteristics.
Geometric Sequences: Definition and Example
- A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant called the common ratio.
- Example: 2, 4, 8, 16, 32, 64, 128 (each term is multiplied by 2).
Finite Geometric Sequence
- Contains a fixed number of terms and has a definite last term.
- Has a first term (a₁) and a last term (aₙ), e.g., 2 (first term), 128 (last term in the example).
- Follows the geometric pattern up to the final term.
Infinite Geometric Sequence
- Continues without end and has no last term.
- Written with three dots (…) to indicate it goes on infinitely, e.g., 2, 4, 8, 16, 32, 64, 128, ...
- Has a common ratio and often a first term, but sometimes no explicit first or last term.
- Can also be written with three dots before and after a segment, showing the sequence extends infinitely in both directions.
Comparing Finite and Infinite Geometric Sequences
- Both types share a common ratio and follow a repeated multiplication pattern.
- Finite sequences have clear starting and ending terms; infinite sequences do not have an end and sometimes do not have a defined beginning.
Key Terms & Definitions
- Geometric Sequence — a sequence where each term is multiplied by a fixed number (the common ratio) to get the next term.
- Common Ratio (r) — the constant factor between consecutive terms (example: r = 2).
- Finite Geometric Sequence — a geometric sequence with a limited number of terms.
- Infinite Geometric Sequence — a geometric sequence that continues indefinitely.
Action Items / Next Steps
- Review the examples of both finite and infinite geometric sequences.
- Practice writing your own examples of each type.