We
have seen that the triangular numbers are formed by the following sequence:
1,
3, 6, 10, 15, 21, É
We
can also represent these in tabular form. WeÕll call the first triangular T(1),
the second triangular number T(2), and so forth.
n 
T(n) 
1 
1 
2 
3 
3 
6 
4 
10 
5 
15 
6 
21 
If
we investigate the finite differences for the T(n) column, weÕll find that the
second order differences are constant:
n 
T(n) 
1^{st}
Differences 
2^{nd}
Differences 
1 
1 


2 
3 
2 

3 
6 
3 
1 
4 
10 
4 
1 
5 
15 
5 
1 
6 
21 
6 
1 
So
how is this useful information to us? Well, it allows to model T(n) as a
quadratic function (since the secondorder differences are constant).
In
other words, T(n) may be expressed in the form an^{2} + bn + c, for
some appropriate values of a, b, and c.
So T(n) = an^{2} +
bn + c
This
implies the following (by substituting the ordered pairs from the table above):
T(1) = a + b + c = 1
T(2) = 4a + 2b + c = 3
T(3) = 9a + 3b + c = 6
We
could use any three of the ordered pairs in the table. Three pairs are
sufficient and necessary, because three equations would allow us to solve for
three unknowns. Normally one chooses the lower ordered pairs for the sake of
convenience, but it is up to the individual to make the decision.
From
here, we recommend the use of a graphing calculator with the ability to
rowreduce a matrix. For our
demonstrations, we are using the TI84 + Silver Edition.
For
those not familiar with matrix operations on the TI84 calculator, we will
demonstrate each part of the process.
First,
create a matrix with the coefficients of a, b, and c, as well as the T(n)
valued associated a, b, and c in each equation above. The matrix we will create
will be the following matrix: .
Press
[2^{nd}] [x^{1}] to access the Matrix menu and arrow to the
right so that EDIT is highlighted. Select Ò1:[A]Ó. Now weÕll edit Matrix A so that it becomes the matrix above.
Type
in the dimensions of the matrix A as 3x4. Next, input the entries into the
matrix. WeÕll solve this system of
equations for a, b, and c by rowreducing matrix A (in Reduced Echelon Form).
After
you have entered the entries for A, quit the matrix menu (by pressing [2^{nd}]
[MODE] to quit).
It
is a good practice to view your matrix on the home calculator screen in order
to verify that it has been entered correctly. Access the matrix menu as we did
above. With the NAMES menu
highlighted, select matrix A by pressing [1] and then press enter.
This
looks like it is supposed to look, so we will proceed. Select the matrix menu
again. Choose [MATH] and arrow
down to the rref( option. Select that one by pressing enter.
Now
go back to the matrix menu (NAMES) and choose Matrix A.
Now
press enter:
So
we can see by matrix operations that a = 0.5, b = 0.5, and c = 0, which implies
the following:
T(n) = 0.5n^{2} +
0.5n + 0 .
So
now we have found a formula for representing the nth triangular number.