Example ① S_n is the sum of the sequence {a_n}. Given S_n = 2a_n − n, find a_n.
Solution:a_{n+1} = S_{n+1} − S_n = 2a_{n+1} − (n+1) − (2a_n − n) → a_{n+1} = 2a_n + 1
Characteristic equation α=2α+1 → α=−1. a_1=S_1=2a_1−1 → a_1=1.
a_n + 1 is geometric with first term 2 and common ratio 2 → a_n = 2ⁿ − 1
Solution:a_{n+1} = S_{n+1} − S_n = 2a_{n+1} − (n+1) − (2a_n − n) → a_{n+1} = 2a_n + 1
Characteristic equation α=2α+1 → α=−1. a_1=S_1=2a_1−1 → a_1=1.
a_n + 1 is geometric with first term 2 and common ratio 2 → a_n = 2ⁿ − 1
Key Point
- The relation a_n = S_n − S_{n−1} (n≧2) appears frequently
Practice Problems
- Find the general term for a_1=1, a_{n+1}=4a_n−6.
Solutions and Explanations
- Solution:a_n = 2 − 4ⁿ⁻¹
Characteristic equation α=4α−6 → α=2. a_n−2 is geometric with common ratio 4 and first term −1.