Sequences

December 7, 2009

The following is a common problem for those getting started working with sequences:

Find a formula for the general term of the sequence:
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, …..
where the number n occurs n times in the sequence as shown.

The first step is to determine when the first occurrence of k will be. It’s not hard to show that this will be $\frac{n^2 - n + 2}{2}$

Call the first occurrence of k $o_k$ . Also, define our sequence as $a_n$ . Then:

For all $o_k \leq i < o_{k+1}$
$a_i = k$

or:

$\frac{n^2 - n + 2}{2} \leq i < \frac{n^2 + n + 2}{2}$

So we want our function for $a_n$ to be some sort of floor or something, such that it maps all of those numbers in the range to n. Now, we see that we can, in a way, invert this function. By this I mean, by inverting our function for $o_k$ . Since for all $o_k \leq i < o_{k+1}$ , $a_i = k$ , by inverting $o_k$ , it will map all of the first occurrence of each integer to the correct value, and then something weird for the other ones. By taking the floor of this, it will put it to the correct value:

$\frac{n^2 - n + 2}{2} = x$