Overview
This lecture covers how to find the nth term of arithmetic sequences, calculate specific terms, analyze
sequence properties, and solve related exam-style questions.
Finding the nth Term of a Sequence
- The nth term of an arithmetic sequence can be written as: nth term = first term + (n - 1) × common difference.
- For increasing sequences, the common difference is added; for decreasing, it is subtracted.
- Example: For 5, 8, 11, 14, ... the nth term = 5 + (n - 1) × 3 = 3n + 2.
Practice Questions: nth Term
- Find the nth term for sequences with both positive and negative common differences (e.g., 9, 14, 19, 24, ... and 10, 7, 4, 1, ...).
- Apply the formula to various sequences, including those with fractional terms.
Calculating Specific Terms
- To find the 100th term, substitute n=100 into the nth term formula for each sequence.
- Compare or sum specific terms (e.g., 10th and 50th, or 100th and 200th terms).
Generating Terms from nth Term
- Use the formula given (e.g., 5n + 3) to find the first 5 terms by substituting n = 1 to 5.
Sequence Membership Questions
- To check if a given number is part of a sequence, set the nth term formula equal to the number and solve for n.
- If n is a positive integer, the number is a term in the sequence.
Sequence Analysis: Properties & Multiples
- Given nth term rules, state whether sequence numbers are always, sometimes, or never multiples of 7.
- Example: nth term 14n is always a multiple of 7.
Example Calculations
- Calculate differences or sums between specific sequence terms using the nth term formula.
- Find which term exceeds a given value by solving nth term > specified value for n.
Key Terms & Definitions
- Sequence — An ordered set of numbers following a specific pattern or rule.
- Arithmetic Sequence — A sequence where each term increases or decreases by a constant amount.
- nth Term — A formula expressing the value of any term in a sequence as a function of n.
- Common Difference — The fixed amount added or subtracted to form the next term in an arithmetic sequence.
Action Items / Next Steps
- Practice finding nth terms for given sequences (Questions 1–2).
- Calculate specific term values for various sequences (Questions 3, 8).
- Analyze sequence properties regarding multiples (Question 4).
- Complete any skipped “Apply” or “Workout” problems as assigned.