Lesson 68: Geometric Sequences and Series
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant called the common ratio (r). A geometric series is the sum of the terms of a geometric sequence.
Key Formulas
- nth term of a geometric sequence: a_n = a_1 × r^(n − 1)
- Sum of first n terms (finite geometric series): S_n = a_1 × (1 − r^n) / (1 − r), r ≠1
- Sum of an infinite geometric series: S_∞ = a_1 / (1 − r), |r| < 1
Examples
Example 1: Find the 6th term of the sequence 3, 6, 12, 24, …
Solution: a_6 = 3 × 2^(6 − 1) = 3 × 32 = 96
Example 2: Find the sum of the first 5 terms of the sequence 2, 4, 8, 16, …
Solution: S_5 = 2 × (1 − 2^5) / (1 − 2) = 2 × (1 − 32) / (−1) = 2 × (−31) / (−1) = 62
Example 3: Find the sum of the infinite geometric series 5, 2.5, 1.25, …
Solution: r = 0.5, |r| < 1 → S_∞ = 5 / (1 − 0.5) = 5 / 0.5 = 10
Practice Problems
- Find the 8th term of 2, 6, 18, …
- Find the sum of the first 6 terms of 3, 9, 27, …
- Find the sum of the infinite series 8, 4, 2, 1, …
- The 5th term of a geometric sequence is 48, and the common ratio is 2. Find the first term.
- Find the sum of the first 10 terms of 1, −3, 9, −27, …