Two-Term Recurrence Relation a_{n+1} = p·a_n + q
Characteristic equation: Find α from α = pα + q. Then
{a_n − α} is a geometric sequence with common ratio p.
Example ① Find the general term for a_1=1, a_{n+1}=2a_n+3.
Solution: Characteristic equation: α=2α+3 → α=−3.
a_n+3 is geometric with first term a_1+3=4 and common ratio 2 → a_n+3 = 4·2ⁿ⁻¹ = 2^{n+1}
a_n = 2^{n+1} − 3
Solution: Characteristic equation: α=2α+3 → α=−3.
a_n+3 is geometric with first term a_1+3=4 and common ratio 2 → a_n+3 = 4·2ⁿ⁻¹ = 2^{n+1}
a_n = 2^{n+1} − 3
Key Point
- The standard pattern is “characteristic equation → shift → geometric sequence”
Practice Problems
- Find the general term for a_1=2, a_{n+1}=3a_n−2.
Solutions and Explanations
- Solution:a_n = 3ⁿ⁻¹ + 1
Characteristic equation α=3α−2 → α=1. a_n−1 is geometric with common ratio 3 and first term 1.