Following is a problem I encountered many times in my high school olympiads, but was never able to solve it. Hence didn’t like it.

What is the 100th term in the sequence ?

Following is a quick solution:

In this post, I will discuss the solution given in The Green Book of Mathematical Problems (problem 14).

Determine a function such that the term of the sequence is given by .

Let’s denote the number of the sequence by , i.e. . The integer first occurs in the sequence when each of the integers from 1 to have already appeared 1 to times, respectively. Hence, if then

for .

Hence we have:

Hence we have . Thus, we have

Now compared to the earlier solution obtained by observing the pattern, one might ask “Is there is a better formula?”. For that, you might also look at the discussion at Math.SE.