triangular numbers?

Answer 1

A simple equation for the Nth triangular number is:

N (N+1) / 2

examples:

1 x 2 / 2 = 1
2 x 3 / 2 = 3
3 x 4 / 2 = 6
4 x 5 / 2 = 10
etc.

You can also use this formula to get the sum of the first N numbers. For example if someone says what is the sum of
numbers from 1 thru 25:

25 x 26 / 2 = 325

numbers from 1 thru 1 million:

1,000,000 x 1,000,0001 / 2 = 500,000,500,000

Answer 2

http://www.nottingham.ac.uk/education/number/gl/triang.html

there is a good graphic on the above site


Triangular Numbers
The first ten triangular numbers are -
1, 3, 6, 10, 15, 21, 28, 36, 45, 55
You can calculate triangular numbers by adding up consecutive numbers. For example, the eighth triangular number is equal to -
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8
- which comes to 36.

As the name suggests, you can visualise triangular numbers as a triangle of points.

Everyone in a group of people shakes hands with everyone else. The total number of handshakes will always be a triangular number. For instance, five people will make ten handshakes.

There is a useful short cut if you want to work out a large triangular number. Suppose you want the 100th triangular number. You could add up all the numbers from 1 to 100. But there is a simpler way. First work out the average number by adding together the first and the last number, and dividing by two -
1 + 100 = 101
101 / 2 = 50.5

Now multiply this average by however many numbers you would have to add up, in this case, by 100 -
50.5 x 100 = 5050
- so the 100th triangular number is 5050.

Answer 3

Numbers of the form 1+2+3+4+...+n=n(n+1)/2 for some n.

Ex: 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, 15, 21, etc

Answer 4

Triangular numbers are all numbers that can be formed of the sum
1+2+3+...+n
so the row begins 1,3,6,10,15,21,......
The name is explained if you for a triangle with dots
first you have one dot
then you add a second row with 2 dots
the thirds row would have 3 and so on.....

Triangular numbers are in the second diagonal of Pascal's Triangle
Tetrahedral numbers, forming layers of a tetrahedron, a three-sided pyramid, like Triangular numbers form the layers of a triangle, are in the third diagonal

Answer 5

particular, each and every of the recommendations are (x,y) = (a million,a million), (4, 3), (15, 8), (fifty 5, 20), (119, 34). This replaced into proved via E.T. Avanesov in 1966. The evidence is basically too long to place up here, yet i'm going to provide a quick caricature. If we enable X - 2x+2, Y = 2y+a million, we get 6y^2 = x^3-4x+6. (*) Factorisation in the cubic field Q(?) = ?^3 -4^? + 6 then ends up in a chain of quartic Thue equations which the author solves via Skolem's p-adic technique. it would be exciting to appreciate if (*) would desire to be solved via the technique of elliptic logarithms.

Answer 6

Triangular Numbers

www.shyamsundergupta.com/triangle.htm

- - - - - - - - - - - -s-

Answer 7

just 4 or it could be 2 and 7 if you think about it

Answer 8

http://www.mathematische-basteleien.de/triangularnumber.htm

Answer 9

http://www.mathematische-basteleien.de/triangularnumber.htm