Applications of Arithmetic Means and Arithmetic Sequences
Three terms a, b, c form an arithmetic sequence ⇔ 2b = a + c (the middle term is the average of the two endpoints)
Example ① When x, 7, 13 form an arithmetic sequence, find x.
Solution:2·7 = x + 13 → x = 1
Solution:2·7 = x + 13 → x = 1
Example ② Among three-digit natural numbers, find the number and sum of those that leave a remainder of 3 when divided by 7.
Solution:Applicable numbers: 101, 108, 115, …, 997. First term 101, common difference 7.
Last term 997 → 101+(n−1)·7=997 → n=129 terms.
Sum = 129·(101+997)/2 = 129·549 = 70821
Solution:Applicable numbers: 101, 108, 115, …, 997. First term 101, common difference 7.
Last term 997 → 101+(n−1)·7=997 → n=129 terms.
Sum = 129·(101+997)/2 = 129·549 = 70821
Key Point
- For integer-condition problems, determine the “first term, common difference, and number of terms” in that order.
Practice Problem
- Find the sum of the multiples of 3 from 1 to 100.
Solution and Explanation
- Answer:1683
3+6+…+99, with 33 terms. 33·(3+99)/2=1683.