Three terms a, b, and c form a geometric sequence (where a, b, and c ≠ 0) ⇔ b² = ac
Example ① Find x such that 2, x, and 8 form a geometric sequence.
Solution:x² = 2·8 = 16 → x = ±4
Solution:x² = 2·8 = 16 → x = ±4
Example ② Find the total amount of principal and interest after 10 years on a principal of 1 million yen at an annual interest rate of 5% (compound interest).
Solution:100 × (1.05)¹⁰ ≒ 100 × 1.6289 = approximately 1.6289 million yen
Solution:100 × (1.05)¹⁰ ≒ 100 × 1.6289 = approximately 1.6289 million yen
Key Point
- Compound interest calculations are a typical application of geometric sequences.
- From b² = ac, x has two solutions, ±.
Practice Problem
- Find x such that 3, x, and 27 form a geometric sequence.
Answer and Explanation
- Answer:±9