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/