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.