# Triangular Numbers Calculator

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

### Triangular Numbers Calculator

## What are triangular numbers?

A triangular number or triangle number counts the objects that can form an equilateral triangle. This concept is not just theoretical; it has practical applications in various fields, from architecture to computer graphics. 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

T_{n}= 1 + 2 + 3 + 4 +...+ (n-2) + (n-1) + n,

where n is a natural number.

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

To prove this formula is true, write the general representation twice and rearrange the terms as shown below.

T_{n} = 1 + 2 + 3 + ...+ (n-2) + (n-1) + n

T_{n} = n + (n-1) + (n-2) +... + 3 + 2 + 1

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T_{n} = (1 + n) + 2 + ( n - 1) + 3 + (n - 2) + ...+ (n-1) + 2 + n + 1

T_{n} = (1 + n) + (1 + n) + (1 + n) +...+ (1 + n) + (1 + n). There are n terms, so

_{n}= n * (n+1) or T

_{n}= n * (n + 1) / 2

## Triangular Numbers Chart 1-100

n | Δ |
---|---|

1 | 1 |

2 | 3 |

3 | 6 |

4 | 10 |

5 | 15 |

6 | 21 |

7 | 28 |

8 | 36 |

9 | 45 |

10 | 55 |

n | Δ |
---|---|

11 | 66 |

12 | 78 |

13 | 91 |

14 | 105 |

15 | 120 |

16 | 136 |

17 | 153 |

18 | 171 |

19 | 190 |

20 | 210 |

n | Δ |
---|---|

21 | 231 |

22 | 253 |

23 | 276 |

24 | 300 |

25 | 325 |

26 | 351 |

27 | 378 |

28 | 406 |

29 | 435 |

30 | 465 |

n | Δ |
---|---|

31 | 496 |

32 | 528 |

33 | 561 |

34 | 595 |

35 | 630 |

36 | 666 |

37 | 703 |

38 | 741 |

39 | 780 |

40 | 820 |

n | Δ |
---|---|

41 | 861 |

42 | 903 |

43 | 946 |

44 | 990 |

45 | 1035 |

46 | 1081 |

47 | 1128 |

48 | 1176 |

49 | 1225 |

50 | 1275 |

n | Δ |
---|---|

51 | 1326 |

52 | 1378 |

53 | 1431 |

54 | 1485 |

55 | 1540 |

56 | 1596 |

57 | 1653 |

58 | 1711 |

59 | 1770 |

60 | 1830 |

n | Δ |
---|---|

61 | 1891 |

62 | 1953 |

63 | 2016 |

64 | 2080 |

65 | 2145 |

66 | 2211 |

67 | 2278 |

68 | 2346 |

69 | 2415 |

70 | 2485 |

n | Δ |
---|---|

71 | 2556 |

72 | 2628 |

73 | 2701 |

74 | 2775 |

75 | 2850 |

76 | 2926 |

77 | 3003 |

78 | 3081 |

79 | 3160 |

80 | 3240 |

n | Δ |
---|---|

81 | 3321 |

82 | 3403 |

83 | 3486 |

84 | 3570 |

85 | 3655 |

86 | 3741 |

87 | 3828 |

88 | 3916 |

89 | 4005 |

90 | 4095 |

n | Δ |
---|---|

91 | 4186 |

92 | 4278 |

93 | 4371 |

94 | 4465 |

95 | 4560 |

96 | 4656 |

97 | 4753 |

98 | 4851 |

99 | 4950 |

100 | 5050 |