An expression that represents a term of the sequence {a_n} using the preceding term is called a recurrence relation.
- a_{n+1} = a_n + d (common difference d) → arithmetic sequence
- a_{n+1} = r·a_n (common ratio r) → geometric sequence
Example ① Find the general term for a_1=3, a_{n+1}=a_n+5.
Solution: Arithmetic sequence, first term 3, common difference 5. a_n = 5n − 2
Solution: Arithmetic sequence, first term 3, common difference 5. a_n = 5n − 2
Example ② Find the general term for a_1=2, a_{n+1}=3a_n.
Solution: Geometric sequence, first term 2, common ratio 3. a_n = 2·3ⁿ⁻¹
Solution: Geometric sequence, first term 2, common ratio 3. a_n = 2·3ⁿ⁻¹
Key Point
- When you see the form, first decide whether it is arithmetic or geometric
Practice Problems
- Find the fifth term when a_1=1 and a_{n+1}=a_n+2.
Solutions and Explanations
- Solution:9
a_n=2n−1, so a_5=9.