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).
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,
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
1 1
121
13 31
1 464 1
1 510 105 1
1 61520156 1
1 72135 35217 1
1 82856 70 56288 1
1 93684 126 126 84369 1
Works Cited
Bogolmony, A. (n.d.).Triangular numbers and gauss summation. Retrieved from http://www.cut-the-knot.org/Curriculum/Algebra/GaussSummation.shtml
Gupta, S. (2002, October 26).Fascinating triangular numbers. Retrieved from http://www.shyamsundergupta.com/triangle.htm
Jovanovic, R. (2002, July 2).Triangular numbers. Retrieved from http://milan.milanovic.org/math/english/triangular/triangular.html
Weisstein, E. (n.d.).Triangular number. Retrieved from http://mathworld.wolfram.com/TriangularNumber.html
Cynthia Thomas Triangular Numbers EMAT 6690 Page 1