Millionth Term

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/