Overview
This lecture introduces arithmetic sequences, explains how to identify them, and demonstrates the use of formulas to find terms and sums within such sequences.
Recognizing Arithmetic Sequences
- An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant.
- The constant difference between terms is called the common difference (d).
- To confirm a sequence is arithmetic, check that each term increases or decreases by the same value.
General Term of an Arithmetic Sequence
- The nth term of an arithmetic sequence is given by: ( a_n = a_1 + (n-1)d ).
- ( a_1 ) represents the first term; d is the common difference; n is the term position.
Finding Terms in the Sequence
- To find any term, substitute the values for ( a_1 ), d, and n into the formula ( a_n ).
- Example: For a sequence 3, 7, 11, 15... the common difference d = 4.
Sum of an Arithmetic Sequence
- The sum of the first n terms (S_n) is given by: ( S_n = \frac{n}{2}(a_1 + a_n) ).
- Alternatively: ( S_n = \frac{n}{2}[2a_1 + (n-1)d] ).
Applications and Examples
- Use these formulas to quickly find the 10th term, sum the first 20 terms, or solve related problems on arithmetic sequences.
Key Terms & Definitions
- Arithmetic Sequence — A sequence of numbers with a constant difference between consecutive terms.
- Common Difference (d) — The fixed value added to each term to get the next term.
- General Term (a_n) — Formula for any term: ( a_n = a_1 + (n-1)d ).
- Sum of Sequence (S_n) — Formula for adding the first n terms: ( S_n = \frac{n}{2}(a_1 + a_n) ) or ( S_n = \frac{n}{2}[2a_1 + (n-1)d] ).
Action Items / Next Steps
- Practice identifying arithmetic sequences and calculating terms using ( a_n = a_1 + (n-1)d ).
- Complete assigned exercises on finding sums with the sequence sum formula.