Triangular numbers are formed by adding together consecutive natural numbers up to and including a given number n. They can be represented geometrically as the number of dots in an equilateral triangle of dots, or algebraically as binomial coefficients in Pascal's triangle. Pascal's triangle contributes to number theory by showing which numbers are triangular and allowing easy calculation of sums of triangular numbers.

1. TRIANGULAR NUMBERS
AND PASCAL

2. TRIANGULAR NUMBERS
• The triangular numbers are formed by partial sum of the series 1 + 2 + 3 + 4 + 5 + 6
+ 7. ... + n (2). In other words, triangular numbers are those counting numbers
that can be written as = 1 + 2 + 3 + ... + n.
[T.sub.1] = 1
[T.sub.2] = 1 + 2=3
[T.sub.3] = 1+2+3=6
[T.sub.4] = 1 + 2 + 3 + 4 = 10
[T.sub.5] = 1 + 2 + 3 + 4 + 5 = 15

3. A triangular number is the number of dots in an equilateral triangle evenly filled with
dots
A triangle number is, equivalently, the sum of the n natural numbers from 1 to n.
.

4. You can express the triangular numbers as binomial coefficients
Pascal's triangle makes a
contribution to many fields
of the number theory.
The red numbers are
... ... triangle numbers.
You even can find the sum
of the triangular numbers
easily.
Example: 1+3+6+10+15=35

5. INTERESTING FINDING
Even and odd triangular numbers
. Theeven triangular numbers in red and odd numbers in
black form pairs in the usual sequence

6. - END -
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