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### Millionth Term of a Sequence

```Date: 06/18/2002 at 19:16:06
From: Lisa Vinson
Subject: word problem w/ sequence

What would be the millionth term in the sequence
1,2,2,3,3,3,4,4,4,4,...?

I don't know where to start!!
```

```
Date: 06/18/2002 at 20:41:25
From: Doctor Ian
Subject: Re: word problem w/ sequence

Hi Lisa,

We can arrange the sequence this way:

1
2 2
3 3 3
4 4 4 4
5 5 5 5 5
6 6 6 6 6 6

How does that help?  Well, let's look at the cumulative number of
terms as a function of row number:

1            1 term total
2 2           3
3 3 3          6
4 4 4 4        10
5 5 5 5 5       15
6 6 6 6 6 6      21

The numbers in the column to the right are called 'triangular
numbers', and there is a formula for finding the nth triangular
number:

n(n + 1)
nth triangular number = --------
2

(Check it against the numbers 1, 3, 6, 10, 15, and 21.)

So what you want to know is:  What is the smallest triangular
number that is less than a million?   Do you see why?

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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