Triangular Numbers
Consider the following arrays of dots:
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Look at the number of dots in each array. They form a sequence of numbers: 1, 4, 9, 16, 25, .... These numbers are usually called square numbers or perfect squares. They are the outcomes when multiplying each natural number by itself.
1 = 1 X 1
4 = 2 X 2
9 = 3 X 3 etc.
It is easy to predict the number of dots in the nth array: n2.
Consider arrays of dots that form triangles:
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The number of dots in each of these arrays also forms a sequence of numbers: 1, 3, 6, 10, 15, .... These numbers are called triangular numbers.
For the square numbers, we were able to identify a formula that said the nth square number is equal to n2. It is also possible to derive a formula for the nth triangular number.
Consider the following arrays of triangular numbers:
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In each array, there are two copies of the triangular numbers, a blue one and a green one. The arrays form rectangles.
The first array is a 1 X 2 rectangle
The second array is a 2 X 3 rectangle
The third array is a 3 X 4 rectangle
It is easy to see then that:
the 1st triangular number = [1 X 2] ÷ 2
the 2nd triangular number = [2 X 3] ÷ 2
the 3rd triangular number = [3 X 4] ÷ 2
the 4th triangular number = [4 X 5] ÷ 2
In general, then
the nth triangular number = [n X (n + 1)] ÷ 2
For example, to find the 10th triangular number, simply substitute 10 for n in the formula:
the 10th triangular number = [10 X 11] ÷ 2 = 55
You can use Triangular Numbers to find a formula for the sum of the 1st n Counting Numbers (1 + 2 + 3 + ... + n = ?)
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email the author: Bruce Jacobs
Last modified: Thursday, February 6, 2003