jmc

Question

For each positive integer n, the mean of the first n terms of a sequence is n. What is the 2008th term of the sequence?

Accepted Answer

Let a_n denote the nth term. Then a_1 + a_2 + + a_20082008 = 2008,so a_1 + a_2 + + a_2008 = 2008^2. Also, a_1 + a_2 + + a_20072007 = 2007,so a_1 + a_2 + + a_2007 = 2007^2. Subtracting these equations, we get a_2008 = 2008^2 - 2007^2 = (2008 + 2007)(2008 - 2007) = 4015.