Overview
This lecture explains how to find the nth term of fractional sequences by analyzing numerators and denominators separately, and demonstrates the method with an example.
Fractional Sequences and Their Structure
- Fractional sequences have terms where both numerators and denominators follow their own sequences.
- Example sequence: 1/2, 3/7, 5/12, 7/17, ...
Finding the nth Term: Numerators
- Look for a pattern in numerators: 1, 3, 5, 7, ...
- The numerators increase by 2 each time (arithmetic sequence).
- General form for arithmetic sequence: nth term = first term + (n − 1) × common difference.
- Here, nth term for numerators is 2n − 1.
Finding the nth Term: Denominators
- Observe denominators: 2, 7, 12, 17, ...
- The denominators increase by 5 each time (arithmetic sequence).
- nth term for denominators is 5n − 3.
Constructing the nth Term of the Fraction
- Combine nth terms to form sequence term: (2n − 1) / (5n − 3).
Using the nth Term Formula
- To find any term in the sequence, substitute n into the formula.
- Example: The 100th term is (2×100 − 1) / (5×100 − 3) = 199/497.
Key Terms & Definitions
- Numerator — the top number in a fraction.
- Denominator — the bottom number in a fraction.
- Arithmetic sequence — a sequence where the difference between terms is constant.
- nth term — a formula to find any specific term in a sequence.
Action Items / Next Steps
- Practice finding nth terms for other fractional sequences.
- Review methods for finding nth terms of linear (arithmetic) sequences.