Triangular Number Sequence (original) (raw)

Last Updated : 23 Jul, 2025

**Triangular Number is a sequence of numbers that can be represented in the form of an equilateral triangle when arranged in a series. The triangular numbers list includes numbers 1, 3, 6, 10, 15... They are a type of figurative numbers.

Representation-of-Triangular-Numbers

First triangular number is **T 1 **= 1.

To obtain the second number, add 2 to T1. Thus the second number becomes 3. Subsequently, to obtain the third number, we add 3 to T2 to arrive at number 6. For the ease of understanding, it can be represented as below:

Triangular-Number-Sequence

Table of Content

**Properties of Triangular Numbers

Please refer Interesting Facts about Triangular Numbers for more such facts.

Triangular Number Formula

We know Tn = 1

We can recursively define other riangular numbers. We can see that we get nth triangular number by adding n to (n-1)the triangular number
Tn = Tn-1 + n

Let us use the above recursive to find the formula by replacing Tn-1 with Tn-2 + (n-1)
Tn = Tn-2 + (n - 1) + n

If we keep doing this, we get
Tn = Tn-3 + (n-2) + (n - 1) + n
......................................................
Tn = 1 + 2 + ............. (n-1) + n

Which is sum of first n natural numbers

Therefore the following formula can be used to calculate the triangular numbers:

**T n = n(n+1)/2

In the above formula, (n+1)/2 is binomial coefficient.

Sum of Triangular Numbers

We can compute the sum as **n*(n+1)*(n+2) / 6

**How does this work?

We mainly need to compute sum of i*(i + 1)/2 from i = 1 to n

Which is equal to 1/2 x Sum(i2) + 1/2xSum(i) where i goes for 1 to n in both sums

We can use formulas of sum of n natural numbers and sum of natural number squares and get the expression as

1/2 x [n x (n+1)/2] + 1/2 x [n x (n + 1) x (2n + 1) / 6[

= n x (n + 1)/4 [ 1 + (2n + 1)/3]

= n x (n + 1)/4[ (2n + 4)/3]

= n x (n + 1) x (n + 2) / 6

How to Find Next Triangular Number from Previous 2

If we are given a sequence of triangular numbers, to determine the next number in the series, follow the below given steps:

Let us find the next triangular number of 6 and 10.

Prime Numbers Whole Numbers
Co-prime Numbers Sequence and Series
Square Numbers Real Numbers

Triangular Number List

Triangular number list has the following numbers:

**1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431...and so on.

Triangular Numbers Solved Examples

**Example 1: Find out 10 th Triangular Number.

**Solution:

To find out T10, we use the formula as mentioned below,

Tn = n(n+1)/2

T10 = 10(10+1)/2

T10 = 10(11)/2

T10 = 55.

**Example 2: The first four triangular numbers are 1,3,6 and 10. Find out the position of number 55 in the sequence.

**Solution:

Here, Tn = 55.

We know that Tn= n(n+1)/2

⇒ n(n+1)/2 = 55

⇒ n2 + n = 55 × 2

⇒ n2 + n -110 = 0

⇒ n2 + 11n - 10n - 110=0

⇒ (n+11)(n-10)=0

⇒ n = -11 or n = 10

Since, a negative number can't be the position in the sequence, therefore n = 10 is a valid solution.

Thus, 55 is at the position 10th

Triangular Number Practice Questions

Try out the following questions on Triangular Numbers.

**Q1. Find the 20th Triangular Number

**Q2. Check if sum of first 10 natural numbers is equal to the tenth triangular number is the list

**Q3. Find the position of 66 in Triangular Number Sequence

**Q4. Find the sum of first five triangular numbers