## Counting Sequence

#### Problem

Consider the infinite sequence:, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, ...

What is the 1000^{th} term?

#### Solution

Without having the nature of the sequence explicitly stated we must make an assumption about its behaviour. It seems reasonable to assume that the sequence is made up of one 1, two 2's, three 3's, and so on.

If the sequence continued up to the last `n`, there would be 1 + 2 + ... + `n` = ½`n`(`n`+1) terms.

We can verify that, ½ × 44 × 45 = 990.

In other words, the 990^{th} term is the last 44, the next 45 terms will be 45, hence the 1000^{th} term must be 45.

What is the `k`^{th} term?

Problem ID: 207 (17 Feb 2005) Difficulty: 2 Star