Find the 276th Letter

Date: 08/15/2001 at 15:06:49
From: Cale McMellon
Subject: Sequence pattern

What is the pattern for this problem? I know the answer is g, but 
isn't there an easier way to find it than writing it out?

Find the letter that is the 276th entry in the following sequence:
g,l,g,l,l,g,l,l,l,g,l,l,l,l,g,l,l,...

Date: 08/15/2001 at 15:37:53
From: Doctor Greenie
Subject: Re: Sequence pattern

Hello, Cale -

Here are the numbers of the terms in the sequence that are g:

    1, 3, 6, 10, 15, ...

In other words, entries 1, 3, 6, 10, 15, ... are g, while the rest 
are l.

These you may recognize as the sums of the first n integers...

    1 = 1
    3 = 1+2
    6 = 1+2+3
   10 = 1+2+3+4
   15 = 1+2+3+4+5

and so on. You may also be familiar with the formula for the sum of 
the first n integers:

          n(n+1)
    sum = ------
             2

Thus, if this expression has the value 276 for some integer n, then 
276 is the sum of the first n integers, and so the 276-th term in your 
sequence will be g. Let's see if this expression has the value 276 for 
some integer n:

    n(n+1)
    ------ = 276
       2

    n^2+n = 552

    n^2+n-552 = 0

    (n+24)(n-23)=0

    n = -24  OR  n = 23

n = -24 doesn't make sense in this problem, since n is the number of 
terms in a sequence. The correct answer is n = 23.

So the sum of the first 23 integers is 276:

    1+2+3+...+22+23 = 276

and, therefore, the 276-th term of your sequence is g.

Please write back if you have any further questions on this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/