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Millionth Term

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Date: 04/16/2001 at 10:25:46
From: Steve
Subject: Millionth Term

What is the millionth term in the following sequence?

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...
```

```
Date: 04/16/2001 at 16:23:52
From: Doctor Rob
Subject: Re: Millionth Term

Thanks for writing to Ask Dr. Math, Steve.

This sequence S(n) is connected to the triangular numbers T(k):

1, 3, 6, 10, ...

by the fact that S(T(k)) = k. In fact, what you need to do to find
S(1,000,000) is to find a value k such that T(k-1) < 1,000,000 <=
T(k), and then k will be your answer.

Now you know that T(k) = k*(k+1)/2, so you have two simultaneous
quadratic inequalities to solve:

k*(k+1) - 2000000 >= 0
k*(k-1) - 2000000 <  0

I would use the quadratic formula to solve each. The first will give
you a lower bound on k; the second will give you an upper bound on k.
The true value of k will be the integer lying between these bounds.

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

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