Triangular Number Sequence

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.

First triangular number is T₁ = 1.

To obtain the second number, add 2 to T_1. 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:

Properties of Triangular Numbers

• The nth triangular number is equal to sum of first n natural numbers, .i.e., n x (n + 1)/2 (see the proof below]
• The sum of two consecutive triangular numbers always gives a perfect square.
• They are well known for their application in solving handshake problems.

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

Triangular Number Formula

We know T_n = 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
T_n = T_n-1 + n

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

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

• Calculate the difference between the two consecutive numbers.
• Increase the difference obtained by 1.
• Determine the next number in the sequence.

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

• Step 1: Calculate the difference between two consecutive numbers. In this case it is 10-6 = 4.
• Step 2: Increment the difference obtained by 1. In our example the difference obtained is 4. thus it becomes 5 after increment.
• Step 3: Next number in the series is thus 10 + 5 = 15.

Related Reads

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,

T_n = 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, T_n = 55.

We know that T_n= 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 10^th

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