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.
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