Triangular Numbers
What numbers are Triangular?
Why are they called Triangular Numbers?
How do you tell if a number is Triangular?
Summary
What Numbers are Triangular?
The first ten Triangular numbers are
1, 3, 6, 10, 15, 21, 28, 36, 45, 55
They are found by using an iterative method
(adding consecutive numbers)
First Triangular number is 0+1=1
Second Triangular number is 1+2=3
Third Triangular number is 3+3=6
Fourth Triangular number is 6 + 4 = 10
Fifth Triangular number is 10 + 5 = 15
What is the Pattern?
Triangular 1 2 3 4 5 6 7 8
Number, Tn
Triangular 1 3 6 10 15 21 28 36
Number
Value
Looking at the Triangular Number, Tn and the Triangular
Number Value, can you see a relationship?
The Next Triangular Number Value is found by taking the
Previous Triangular Number Value and adding the next
Triangular Number Tn.
What is the ninth Triangular Number, T9?
Answer: T9 = 36 + 9 = 45
Why are They Called
Triangular Numbers?
They are called Triangular
Numbers because of the number of
dots required to create a triangle.
Compare the Pattern with the pictures.
What do you think?
How do You Tell if a Number is
Triangular?
Consider the image below:
The drawing shows the 3rd triangular number twice,
or 2T3 symbolically.
It also shows a 3x3 square with three extra dots.
How does this Help Find the 75th
Triangular Number?
The area of the drawing is 4x3 = 12
Given that 2T3 = 12 = 3x3 + 3, then the 3rd
Triangular Number, T3 is 12/2 = 6.
To find the 75th Triangular Number, we need to
have a formula for computing every Triangular
Number.
The 75th Triangular Number
1st Triangular Number = (1x2)/2 = 1
2nd Triangular Number = (2x3)/2 = 3
3rd Triangular Number = (3x4)/2 = 6
6th Triangular Number = (6x7)/2 = 21
nth Triangular Number = n(n + 1)/2 o oooooo
oo ooooo
ooo oooo
oooo ooo
ooooo oo
Thus, T75 = 75(75 + 1)/2 = 75(76)/2 = 2850 oooooo o
2T6
Summary
A Triangular Number, Tn, is any number
that can be arranged in the form of a
triangle.
Triangular Numbers can be found using
an iterative method: 1+2+3+4+…+n
There is a formula for finding every
Triangular Number,
Tn = n(n + 1)/2