Arithmetic Sequences: Comprehensive Exercises
Example ① Find the value of n for which the sum S_n of the arithmetic sequence {a_n} is greatest. a_1=30, d=−4.
Solution:a_n = 30−4(n−1) = 34−4n。a_n ≧ 0 ⇔ n ≦ 8.5。
The sum is greatest when adding terms up to n = 8。The maximum occurs when n = 8（S_8 = 8·(30+6)/2 = 144）
Solution:a_n = 30−4(n−1) = 34−4n。a_n ≧ 0 ⇔ n ≦ 8.5。
The sum is greatest when adding terms up to n = 8。The maximum occurs when n = 8（S_8 = 8·(30+6)/2 = 144）
Key Point
- The sum is greatest ⇔ before adding negative terms
Practice Problem
- In an arithmetic sequence with first term 20 and common difference −3, find the greatest n such that a_n > 0。
Solution and Explanation
- Answer:n = 7
20−3(n−1)>0 → n<23/3 ≒ 7.67。