Problem: The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning with 2, is:
Answer Choices:
A. 27
B. 2741​
C. 2721​
D. 28
E. 2821​
Solution:
Method I. We have an arithmetic sequence with the first term a=2, the common difference d=1, and the last term l=a+(n−1)d=2+(52−1)(1)=53. Since S52​=252​(2+53), AM. =5252/2(2+53)​=255​=2721​.
Method II. Designate the terms of the arithmetic sequence by u1​,u2​,…,un​. Then A.M. =21​(u1​+un​) since 21​(u1​+u2​)=21​(a+a+(n−1)d= 21​⋅nn(2a+(n−1)d)​=21​⋅nSn​​.∴A.M.=21​(55)=2721​.
Comment. Generally, A. M. =2ui​+un+1−i​​,i=1,2,…,n.