The general term of an arithmetic sequence with first term a and common difference d is
a_n = a + (n − 1)d
Example ① Find the 10th term of the arithmetic sequence with first term 5 and common difference 3.
Solution:a_10 = 5 + (10−1)·3 = 5 + 27 = 32
Solution:a_10 = 5 + (10−1)·3 = 5 + 27 = 32
Example ② Find the general term of an arithmetic sequence whose third term is 11 and seventh term is 27.
Solution:a+2d=11、a+6d=27 → 4d=16、d=4、a=3。
a_n = 3 + 4(n−1) = 4n − 1
Solution:a+2d=11、a+6d=27 → 4d=16、d=4、a=3。
a_n = 3 + 4(n−1) = 4n − 1
Key Points
- “First term + (n−1) × common difference” is the essence of the formula
- Determine a and d by solving the system of two equations
Practice Problem
- Find the 20th term of the arithmetic sequence with first term 2 and common difference −3.
Answer and Explanation
- Answer:−55
2 + 19·(−3) = 2 − 57 = −55.