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Triangular numbers when arranged in a series or sequence of equilateral triangles represent a sequence where the sum of previous number and order of succeeding numbers results in a sequence of triangular numbers. 1, 3, 6, 10, 15, 21, 28, 36, 45, etc is the sequence of such numbers. When the previous number in a series is added to the order of the succeeding number, a triangular number sequence is generated.
Also read: Isosceles Triangle Theorems
Key Terms: Triangle, Triangular Number, Sum, Sequence, Number, Figurative Number
Triangular Number: Definition
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A triangular number (Tn ) is defined as a figurative number which can be represented in the form of an equilateral triangle consisting of several rows of elements such that every subsequent row contains one element more than the previous row.
Triangular Number
The above figure represents different triangular number sequences. For instance, T1= 1 represents a triangular number consisting of a single element. On the other hand, T4= 10 represents a triangular number sequence consisting of elements with each row having one element more than the previous row. Thus 1, 1+2, 1+2+3, 1+2+3+4,..etc for n= 1,2,3, … are the first few triangular numbers.
Where is a binomial coefficient. In simple words, the triangular number, n + (n-1) + (n-2) + ……+ 2+ 1 is an additive analog of the factorial n! = n.(n-1).(n-2)……2.1.
Also Read: Pair of Linear Equations in Two Variables Formula
Triangular Number List
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As already stated, a triangular number sequence is represented by a row of elements where every subsequent row contains one element more than the previous row. Thus the triangular number list consists of the following elements:
0, 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.
Sum of Triangular Numbers
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In a triangular number sequence, the next number is added with an extra row.
This is explained in detail in the following:
The first number is 1.
To the first number two dots are added.
In number 3, one more dot is added to the existing two dots.
In number 4, a row of four dots is added and so on.
So the sequence of triangular numbers is as follows.
1, 1+2, 1+2+3, 1+2+3+4, and so on.
Also Read: Factorial Formula
Determination of Triangular Numbers
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A triangular number ‘N’ is represented by the sum of all "n" natural numbers from "1" to "n" with the number of black dots in a triangular pattern with n black dots on each side.
Also Read: Cosine Rule
Relationship of Triangular Numbers with other Figurative Numbers
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There is a broad relationship between triangular numbers and other figurative numbers. Thus when two triangular numbers are added together it results in a square value.
For instance, 1, 36, 1225, etc are some of the triangular numbers which also have square values.
Roots of Triangular Number and Test of Triangular Number
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In similar lines to the square root of x, the positive triangular root of x can be defined as the number ‘n’ where n = √8x+1−1/2 and thus Tn = x. This result can be concluded from the quadratic formula very easily. Thus it can be concluded that a square number 8x + 1 is triangular if and only if x is triangular.
It is to be noted that the difference between the nth m-gon number and the nth (m + 1)-gon number is the (n−1) th triangle number. For instance, the difference between the sixth heptagonal number and the sixth hexagonal number is the fifth triangular number. Apart from that every other triangle number is a hexagonal number. Lastly it should be remembered that the difference between the two triangular numbers is a trapezoidal number.
Read More: Angle of Depression
Properties of a Triangular Number
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- Triangular numbers are correlated to the first-degree case of Faulhaber’s formula.
- Every alternate triangular number (1, 6, 15, 28, 45….) is also a hexagonal number.
- For base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Thus every triangular number is either divisible by 3 or leaves a remainder of 1 when divided by 9.
- Triangular numbers aren’t divisible by 3 and thus when divided by 27, they either have to leave a remainder of 1 or 10.
- Every nine terms are repeated in a digital root pattern of triangular numbers. Example "1, 3, 6, 1, 6, 3, 1, 9, 9". But the reverse is not true. For instance, the digital root of 12(not a triangular number), is 3 and divisible by three.
Read More: Inverse Trigonometric Formulas
Application of Triangular Number
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In order to design a fully connected network of n computing devices, Tn − 1 cables or other connections are required.
A tournament using round-robin group stage, the number of matches that are required to be played between n teams is equal to the triangular number Tn – 1.
Depreciation of an asset can be assessed using triangular numbers wherein the sum of years' digit method is used, which involves finding Tn, where n is the length in years of the asset's useful life.
To determine height or distance between objects in trigonometry, triangular numbers are widely used.
Triangular numbers are also used in civil engineering design of buildings etc.
Also Read:
Things to Remember
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- Triangular numbers are figurative numbers arranged in a sequence where every subsequent row contains one element more than the preceding row.
- 1, 3, 6, 10, 15, 21, 28 etc represent a triangular number sequence.
- The above formula is used to represent a triangular number.
- All square triangular numbers are available from recursion.
- The square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n.
- The difference between the nth m-gonal number and the nth (m + 1)-gonal number is the (n − 1)th triangular number.
- The positive difference of two triangular numbers is a trapezoidal number
- All perfect numbers are triangular.
Sample Questions
Ques. Mention different ways of representing triangular numbers? (2 Marks)
Ans: There are four major ways of representing triangular numbers which are listed below:
- by the sum Tn=∑ni=1iTn=∑i=1ni;
- by the recurrence relation T1=1T1=1, Tn=Tn−1+nTn=Tn−1+n;
- by the closed formula Tn=n(n+1)2Tn=n(n+1)2;
- by the binomial coefficient Tn=(n+12)Tn=(n+12).
Ques. Are all perfect numbers being treated as triangular? (2 Marks)
Ans: All even perfect numbers can be treated as triangular. Since every even perfect number can be represented as
where Mp=2p−1Mp=2p−1 is a prime. Ques. What is the largest triangular number? (1 Marks)
Ans: 4095 is the largest triangular number of the form 2k − 1.
Ques. How can the handshaking problem be resolved using a triangular number? (2 Marks)
Ans: The handshake problem deals with counting the number of handshakes if each person in a room shakes hands once with n + 1 people in the same room. The solution to the above problem is Tn−1 where the function T is the additive analog of the factorial function, which is the product of integers from 1 to n.
Ques. How do you find the difference between two triangular numbers? (2 Marks)
Ans: The difference between two consecutive triangular numbers is a natural number series consisting of elements 1, 2, 3, 4, 5, 6, and so on. Thus the nth triangular number is obtained by adding “n” to the (n-1)th triangular number.
Ques: What is the area of an equilateral triangle whose perimeter is 15 cm. (2 marks)
Ans. The perimeter of an equilateral triangle is equal to side *3.
Here the perimeter is given as 15 cm.
So the side measures 5 cm each.
Now the area of an equilateral triangle is equal to (\(?3\)/4) (side)2.
So area of an equilateral triangle = 25(\(?3\)/4) cm2.
Ques: Find the area of a triangle with all sides equal and where each side measures 16 cm. (2 marks)
Ans. A triangle, whose all sides are equal to one another, is called an equilateral triangle. The area of an equilateral triangle is equal to (√3/4) (side)2 .
Area= √3/4 *16*16
= 64√3 cm2.
Ques. \(\pi \)is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is π = c/ d. This seems to contradict the fact that n is irrational. How will you resolve this contradiction? (2 marks)
Ans. When we measure the length of a line with a scale or with any other device, we only get an approximate rational value, i.e. c and d both are irrational.
∴ c/ d is irrational and hence π is irrational.
Thus, there is no contradiction in saying that it is irrational.
Ques. Is zero a rational number? Can you write it in the form p/ q , where p and q are integers and q ≠0? (2 marks)
Ans. Yes, zero is a rational number; it can be written in the form p/ q.
0 = 0/1 = 0/ 2 = 0/ 3 etc.
Denominator q can also be taken as a negative integer.
Ques: Express 0.66666…… in the form p/q, where p and q are integers and q ≠ 0. (2 marks)
Ans: 0.66666……
Let, x = 0.66666……
10x = 6.666….
10x – x = (6.6666…..) – (0.66666……..)
9x = 6
x = 6/9
x = 2/3
Thus, 0.6666… is 2/3 in p/q form.
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