TRIANGULAR NUMBERS ARE EVERYWHERE!

by: Charles Hamberg
Illinois Mathematics and Science Academy

The sequence 1, 3, 6, 10, 15, …, n(n + 1)/2, … shows up in many places of mathematics. To the Greeks these numbers were known as the triangular numbers due to the association with the triangular array of dots:

one row of one dot a row of one and a row of two row of one, row of two, and a row of three rows of one, two, three, and four dots rows of one, two, three, four, and five dots
1 = 1 3 = 1 + 2 6 = 1 + 2 + 3 10 = 1 + 2 + 3 + 4 15 = 1 + 2 + 3 + 4 + 5

We observe that the triangular numbers can also be associated with the sums of consecutive natural numbers beginning with 1.

If we let Tn = 1 + 2 + 3 + … + n we can find a closed form for Tn. For convenience, T0 is defined to be 0. Then:

Tn = 1 + 2 + 3 + … + (n - 1) + n
Tn = n + (n - 1) + (n - 2) + … + 2 + 1
Adding gives 2Tn = (n + 1) + (n + 1) + (n + 1) + … + (n + 1) + (n + 1)
There are n groups of (n + 1), so we see that 2Tn = n(n + 1) or Tn = n(n + 1)/2.

We shall now look at several examples of how triangular numbers appear in mathematical settings.

Example 1: Complete the table :
single point
line segment AB
line segment through points A, B, and C
line segment through points A, B, C, and D
line segment through points A, B, C, D, and E
Number of points 1 2 3 4 5 6 7 n
Number of line segments
connecting pairs of points 		0 1 3 6 10 ? ? ?

Example 2: Complete the table:
a ray
two rays with common endpoint
three rays with common endpoint
four rays with common endpoint
Number of rays 1 2 3 4 5 6 n
Number of angles 0 1 3 6 ? ? ?

Example 3:
number of terms number of non-square terms
(a)2 = a2 1 0
(a + b)2 = a2 + b2 + 2ab 3 1
(a + b + c)2 = a2 + b2 + c2 +
2ab + 2ac + 2bc 		6 3
(a + b + c + d)2 = ? ? ?
(a + b + c + d + e)2 = ? ? ?
(a1 + a2 + … + an)2 = ? ? ?

Example 4: Complete the table:
triangle quadrilateral drawn with its diagonals number of sides
of polygon 		number of
diagonals
3 0 = 1 - 1 = 0 + 0
4 2 = 3 - 1 = 1 + 1
0 diagonals 2 diagonals 5 5 = 6 - 1 = 3 + 2
6 9 = 10 - 1 = 6 + 3
pentagon drawn with its diagonals hexagon and its diagonals 7 ? = ? - ? = ? + ?
8 ? = ? - ? = ? + ?
: :
: :
5 diagonals 9 diagonals n ? = ? - ? = ? + ?

Example 5: Complete the chart:
a straight line two intersecting lines number of straight lines number of angles formed (<180°)
1 0 = 4·0
2 4 = 4·1
three concurrent lines four concurrent lines 3 12 = 4·3
4 24 = 4·6
5 ? = ?
6 ? = ?
n ? = ?

Exploration 1:
(a) Find the sum of 1 + 2 + 3 + . . . + 500
(b) Find the sum of 100 + 101 + 102 + . . . + 500
(c) Find the sum of 2 + 4 + 6 + 8 + . . . + 2n
(d) Find the sum of 1 + 3 + 5 + . . . + (2n - 1)
(e) Find the sum of 1 + 3 + 5 + . . . + 99
(f) Find the sum of 47 + 49 + 51 + . . . + 99

Exploration 2: Determine the number of rectangles in each checkerboard.
(a) 1 x 1 checkerboard, 2 x 2 checkerboard, 3 x 3 checkerboard, 4 x 4 checkerboard
(b) 1 x 2 checkerboard, 2 x 3 checkerboard, 3 x 4 checkerboard, 4 x 5 checkerboard

Exploration 3:
(a) Count the rectangles in each: strips of 1, 2, 3, 4, 5 squares
(b) Count the triangles in each: copies of a triangle with 0, 1, 2, 3 cevians (segments joining a vertex with the opposite side) drawn from the same vertex
(c) Count the triangles in each: copies of two triangles stuck base to base, the top one having 0, 1, 2, 3 cevians drawn from vertex to common base, the bottom one having 1, 2, 3, 4 cevians drawn from vertex to common base

Exploration 4: Find the sum of all the numbers in the triangular array of numbers:

first row: 1; second row: 2, 3; third row: 4, 5, 6; ...; ninth row 37, 38, ..., 44, 45

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