A sequence in which the ratio of any two consecutive terms is constant is called a geometric sequence, and this ratio r is called the common ratio.
a_{n+1}/a_n = r (constant)
General term of a geometric sequence with first term a and common ratio r:
a_n = a·rⁿ⁻¹
Example ① Is 3, 6, 12, 24, … a geometric sequence? Find its general term.
Solution: Since the ratio is always 2, it is a geometric sequence. a_n = 3·2ⁿ⁻¹
Solution: Since the ratio is always 2, it is a geometric sequence. a_n = 3·2ⁿ⁻¹
Key Point
- Geometric sequence ⇔ the ratio of consecutive terms is constant
- The common ratio may be negative or fractional
Practice Problems
- Find the 6th term of the geometric sequence with first term 5 and common ratio 2.
Answer and Explanation
- Answer: 160
5·2⁵=160.