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"What are Triangular Numbers?"

Simply Maths


Triangular numbers are a sequence of numbers that can be represented as equilateral triangles.

triangular numbers

The nth triangular number is the sum of all positive integers from 1 to n:


1st number (n = 1) = 1

2nd number (n = 2) = 1 + 2 = 3

3rd number (n = 3) = 1 + 2 + 3 = 6

4th number (n = 4) = 1 + 2 + 3 + 4 = 10 ... and so on.


There is a pattern so we should be able to get a formula to work them out. We know in a rectangle that the area can be found by multiplying the length by the height.


If we draw the dots like so ...

trangle of dots

... and put two similar figures together ...

rectangle of dots

... a rectangle is formed. If you do this for any of the triangular numbers, the base is always 1 more than the height. Taking the height to equal n, the base will be n + 1. The number of dots in the rectangle will be n x (n + 1). The triangular number, T(n), will be half this number and can now be calculated using the following formula:

T(n) formula

Let's break down how to calculate a triangular number step by step:

  1. Determine which triangular number you want to calculate (n).

  2. Plug the value of n into the formula: T(n) = n(n + 1) / 2

  3. Multiply n by (n + 1)

  4. Divide the result by 2:

Let's calculate a few examples:


Example 1: Calculate the 4th triangular number (T(4)).

T(4) = 4 x (4 + 1) / 2 = (4 x 5) / 2 = 20 / 2 = 10

So, the 4th triangular number is 10.


Example 2: Calculate the 7th triangular number (T(7)).

T(7) = 7 x (7 + 1) / 2 = 7 x 8 / 2 = 56 / 2 = 28

So, the 7th triangular number is 28.


Example 3: Calculate the 20th triangular number (T(20)).

T(20) = 20 x (20 + 1) / 2 = 20 x 21 / 2 = 420 / 2 = 210

So, the 20th triangular number is 210.


You can use this method to calculate any triangular number by plugging in the value of n into the formula and performing the calculations step by step.


Triangular Numbers - Facts and Properties:


Triangular numbers...


,,, can be represented geometrically as equilateral triangles. The nth triangular number corresponds to the number of dots in a triangular arrangement, where each row contains one more dot than the previous row.


... are the sums of consecutive positive integers. For example, the 4th triangular number (T(4) = 10) is the sum of 1 + 2 + 3 + 4.


... can be thought of as a special case of arithmetic progressions, where the common difference between terms is 1.


... produce appealing patterns when represented as triangles. For example, you can stack triangular numbers to form larger equilateral triangles, and these patterns have been used in art and design.


... can be expressed as sums of consecutive square numbers. For example, T(4) = 1^2 + 2^2 = 5.


... appear in various combinatorial problems, such as counting the number of handshakes in a group of people or the number of ways to arrange objects in a triangular grid.


... are related to Pascal's Triangle, a famous triangular array of binomial coefficients. The sum of the numbers in the nth row of Pascal's Triangle is the nth triangular number.


... can be found in natural patterns, such as the arrangement of petals on flowers, the stacking of oranges in a crate, or the growth patterns of certain plant leaves.


... can now be calculated by you!


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