# Triangular Numbers Calculator

Here is the answer to questions like: 19 triangular number or what is 19 triangular number?

Use the triangular numbers tool below to calculate the triangular number of any given number. Find below in this web page a triangular numbers list from 1 to 100 as well as the nth term formula for triangular numbers with its demonstration.

## What are triangular numbers?

A triangular number or triangle number counts the objects that can form an equilateral triangle. The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n.

The general representation of a triangular number is

Tn= 1 + 2 + 3 + 4 +...+ (n-2) + (n-1) + n,

where n is a natural number.

This sum is Tn = n * (n + 1) / 2. This is the triangular number formula to find the nth triagular number.

To prove that this formula is true, write twice the general representation and rearange the terms as below

Tn = 1 + 2 + 3 + ...+ (n-2) + (n-1) + n
Tn = n + (n-1) + (n-2) +... + 3 + 2 + 1
---------------------------------------
Tn = (1 + n) + 2 + ( n - 1) + 3 + (n - 2) + ...+ (n-1) + 2 + n + 1
Tn = (1 + n) + (1 + n) + (1 + n) +...+ (1 + n) + (1 + n). There are n terms, so

2Tn = n * (n+1) or Tn = n * (n + 1) / 2

nΔ
11
23
36
410
515
621
728
836
945
1055
nΔ
1166
1278
1391
14105
15120
16136
17153
18171
19190
20210
nΔ
21231
22253
23276
24300
25325
26351
27378
28406
29435
30465
nΔ
31496
32528
33561
34595
35630
36666
37703
38741
39780
40820
nΔ
41861
42903
43946
44990
451035
461081
471128
481176
491225
501275
nΔ
511326
521378
531431
541485
551540
561596
571653
581711
591770
601830
nΔ
611891
621953
632016
642080
652145
662211
672278
682346
692415
702485
nΔ
712556
722628
732701
742775
752850
762926
773003
783081
793160
803240
nΔ
813321
823403
833486
843570
853655
863741
873828
883916
894005
904095
nΔ
914186
924278
934371
944465
954560
964656
974753
984851
994950
1005050