Lesson 67: Arithmetic Sequences and Series
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d). An arithmetic series is the sum of the terms in an arithmetic sequence.
Key Formulas
- nth term of an arithmetic sequence: a_n = a_1 + (n − 1)d
- Sum of first n terms (arithmetic series): S_n = n/2 × (a_1 + a_n) or S_n = n/2 × [2a_1 + (n − 1)d]
Examples
Example 1: Find the 10th term of the sequence 3, 7, 11, 15, …
Solution: a_10 = 3 + (10 − 1)·4 = 3 + 36 = 39
Example 2: Find the sum of the first 15 terms of the sequence 2, 5, 8, 11, …
Solution: a_15 = 2 + (15 − 1)·3 = 2 + 42 = 44
S_15 = 15/2 × (2 + 44) = 7.5 × 46 = 345
Practice Problems
- Find the 20th term of the sequence 5, 9, 13, …
- Find the sum of the first 12 terms of 7, 10, 13, …
- The 8th term of an arithmetic sequence is 22, and the 15th term is 50. Find a_1 and d.
- Find the sum of the first 25 terms of the sequence 2, 6, 10, …
- Determine the 30th term of the sequence −3, 0, 3, 6, …