Sum of an arithmetic sequence with first term a, last term l, and number of terms n:
S_n = n(a + l)/2
When the first term is a, the common difference is d, and the number of terms is n:
S_n = n{2a + (n−1)d}/2
Example ① The sum of 1 + 3 + 5 + … + 99.
Solution: The first term is 1, the last term is 99, and the number of terms is 50.
S = 50·(1+99)/2 = 2500
Solution: The first term is 1, the last term is 99, and the number of terms is 50.
S = 50·(1+99)/2 = 2500
Key Point
- “Number of terms × (first term + last term) ÷ 2” is the shortest route.
Practice Problems
- Find the sum up to the 20th term of an arithmetic sequence with first term 5 and common difference 4.
Solution and Explanation
- Answer: 860
20·{10+19·4}/2 = 10·86 = 860.