Triangular Numbers
Triangular numbers are figurate numbers that can be represented by an
arrangement of dots in a triangular shape. Figurate numbers are the broader idea of
any number that can be represented by dots in a geometric shape. The following
graphic displays the triangular numbers 3, 6, and 10:
One is also considered a triangular number. Higher triangular numbers can be
found by adding a new row to a triangle that is one dot larger than the previous largest
row. Hence, the next triangular number would be fourteen, found by adding a row of
four to the largest triangle above.
The triangular numbers can also be represented as a summation. If we consider
the first triangular number t1 = 1, and the second triangular number t2 = 3, and so on,
then the triangular numbers can be found by adding all of the positive numbers which
are less than or equal to the value of k for tk. This implies that a triangular number is
equal to
∑k from k = 1 to n.
If we are looking for the 5th triangular number (t5), k = 5,
t5 = 5 + 4 + 3 + 2 + 1 = 15.
From the summation formula it is easy to see that the triangular numbers are
equivalent to:
½ * k (k+1).
Cynthia Thomas
Triangular Numbers
EMAT 6690
Page 1
Since we already know the 5th triangular number we can observe this new formula for
triangular numbers:
½ * (5) * (5+1) = ½ * 5 * 6 = 15, the same result as was found
with the summation formula.
Gauss was the first known mathematician to use this formula (inadvertently). As
the story is told, as a ten year old child Gauss’ teacher asked his class to sum up the
numbers from one to one hundred. Gauss wrote down the (correct) answer 5050 on his
paper without doing any work. When asked how he came up with this answer he said
he noticed if you sum the first and last numbers, 1 and 100, you get 101. Then if you
sum the second and second-to-last numbers, 2 and 99, you get 101. So he saw that
you would have 101 fifty times, since 100/2 = 50. Thus you have,
101*50 = 5050.
Therefore the sum of numbers from 1 to n is (n/2)(n+1), where n/2 is the number of
pairs (50 for the problem), and n+1 is the sum of each pair (101). This formula is the
same as the formula for finding the nth triangular number.
This formula makes it easy to make other observations about the triangular
numbers. Pentagonal numbers are
figurative numbers that form
pentagons, such as 1, 5, and 12 (left).
The pentagonal numbers can be
defined in terms of the triangular
numbers. Every pentagonal number
is one third of a triangular number.
The first pentagonal number (1) is one
third of the second triangular number
(3). The second pentagonal number (5) is one third of the fifth triangular number. This
pattern continues to give for a pentagonal number, pi, where i is the order in which the
pentagonal number falls,
pi =(1/3) t3i-1.
The hexagonal numbers, numbers that create hexagons, can be defined similarly
in terms of the triangular numbers. Every other triangular number is a hexagonal
number, or:
hi =t2i-1.
The square numbers are also related to the triangular numbers. The sum of any
two consecutive triangular numbers is a square number. So, for two consecutive
triangular numbers tn and tn-1,
Cynthia Thomas
Triangular Numbers
EMAT 6690
Page 2
tn + tn-1 = ½ *k*(k+1) + ½ *(k-1)(k)
= ½ (k2+k) + ½ *(k2-k)
= ½ (2k2)
= k2.
There are many other interesting facts about triangular numbers, including but
not limited to:





All even perfect numbers are triangular tp where p is a prime number.
The only Fibonacci numbers which are triangular are 1, 3, 21, and 55.
The only Pell number that is triangular is 1.
666 is a triangular number.
The only triangular numbers that are the product of three consecutive integers
are 6, 120, 210, 990, 185136, 258474216.
 Part of Fermat’s Polygonal Theorem states that every positive integer is the sum
of at most three triangular numbers (proven by Gauss in 1796).
 Adding up consecutive triangular numbers produces the tetrahedral numbers.
 The triangular numbers can be found on the third diagonal of Pascal’s triangle,
beginning at row three (below).
1
11
121
1331
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
Cynthia Thomas
Triangular Numbers
EMAT 6690
Page 3
Works Cited
Bogol mony, A. ( n.d.) . Triangular numbe rs and gauss summation . Re tri e ve d from http: / /w ww .cut- the knot.org/ Curri cul um/ Al ge bra/ GaussSummati on.shtml
Gupta, S. ( 2002, O ctobe r 26) . Fascinating triangular numbe rs . Re trie ve d from
http: / /ww w .shyamsunde rgupta.com/ triangle .htm
J ovanovi c, R. ( 2002, J ul y 2) . Triangular numbe rs . Re tri e ve d from
http: / / mi lan.mil anovi c.org/ math/e ngli sh/ triangular/ tri angul ar.html
We i sstei n, E. ( n.d.) . Triangular numbe r . Re tri e ve d from
http: / / mathw orl d.w ol fram.com/ Tri angul arNumbe r.html
Cynthia Thomas
Triangular Numbers
EMAT 6690
Page 4