How to Find the nth Triangular Number: Formula and Examples
The concept of triangular numbers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Triangular numbers are commonly encountered in pattern recognition, number theory, and visual mathematics, and understanding this sequence can help you quickly solve sequence-based questions in exams.
What Is Triangular Numbers?
A triangular number is defined as a number that can be represented as a triangle formed by arranging dots in successive rows, each row having one more dot than the previous. For example, the first few triangular numbers are 1, 3, 6, 10, 15, 21, and so on. You’ll find this concept applied in areas such as arithmetic progressions, combinatorics, and real-world arrangements.
Key Formula for Triangular Numbers
Here’s the standard formula: \( T_n = \frac{n(n+1)}{2} \), where \( n \) is a natural number. This formula gives the nth triangular number efficiently.
Cross-Disciplinary Usage
Triangular numbers are not only useful in Maths but also play an important role in Physics (e.g., network connections), Computer Science (e.g., handshake problems), and logical reasoning. Students preparing for JEE or NEET will see their relevance in various questions, especially those involving patterns or arrangements.
Step-by-Step Illustration
- Suppose you want the 6th triangular number.
Use the formula: \( T_6 = \frac{6 \times 7}{2} \)
- Multiply: 6 × 7 = 42
Now divide by 2: 42 ÷ 2 = 21
- So, the 6th triangular number is 21.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for checking if a number is a triangular number. Many students use this trick during timed exams to save crucial seconds.
Example Trick: To check if 36 is a triangular number, use this method:
- Multiply the number by 8 and add 1: (8 × 36) + 1 = 289
- Take the square root of 289: √289 = 17
- If you get a whole number, the original number is triangular. So, 36 is a triangular number (it is the 8th in the sequence).
Tricks like this are not just cool—you’ll find them practical in competitive exams like NTSE, Olympiads, and even JEE. Vedantu’s live sessions include more such shortcuts to help you build speed and accuracy.
Triangular Numbers List (First 20)
| n | Tn |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 6 |
| 4 | 10 |
| 5 | 15 |
| 6 | 21 |
| 7 | 28 |
| 8 | 36 |
| 9 | 45 |
| 10 | 55 |
| 11 | 66 |
| 12 | 78 |
| 13 | 91 |
| 14 | 105 |
| 15 | 120 |
| 16 | 136 |
| 17 | 153 |
| 18 | 171 |
| 19 | 190 |
| 20 | 210 |
Frequent Errors and Misunderstandings
- Assuming triangular numbers are the same as square numbers.
- Incorrect formula usage by forgetting to divide by 2.
- Counting mistake when adding up numbers in sequence.
Relation to Other Concepts
The idea of triangular numbers connects closely with topics such as arithmetic sequences and Pascal's Triangle. Mastering this helps with number patterns, combinatorics, and advanced topics like square numbers.
Try These Yourself
- Write the first five triangular numbers.
- Check if 48 is a triangular number.
- Find all triangular numbers between 30 and 60.
- Identify non-triangular numbers from the list: 12, 15, 18.
Classroom Tip
A quick way to remember triangular numbers is to visualize arranging balls or dots in equal rows, forming a triangle. Vedantu’s teachers often use diagrams and dot games to help you spot the pattern quickly and make learning interactive.
We explored triangular numbers—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept. For more on number patterns and easy tricks, visit our Maths section and try out more challenging problems yourself!
FAQs on Triangular Numbers Explained with Definition, Formula & Sequence
1. What is a triangular number in Maths?
In mathematics, a triangular number is a figurate number that can be represented as a triangle of dots. It's the sum of consecutive natural numbers starting from 1. The first few triangular numbers are 1, 3, 6, 10, 15, and so on.
2. How do you find the nth triangular number?
The nth triangular number (Tn) can be calculated using the formula: Tn = n(n+1)/2. This formula directly gives you the sum of the first n natural numbers. For example, the 5th triangular number is 5(5+1)/2 = 15.
3. Why are 1, 3, 6, and 10 called triangular numbers?
These numbers are called triangular because they can be arranged visually as equilateral triangles using dots. 1 forms a single dot, 3 forms a small triangle, 6 a larger one, and so on. Each number represents the total number of dots in its corresponding triangle.
4. Is 28 a triangular number?
To check if 28 is a triangular number, we use the formula. We need to solve n(n+1)/2 = 28 for n. This gives n² + n - 56 = 0. Solving this quadratic equation yields n = 7. Since we get a positive integer solution, 28 is indeed a triangular number.
5. What is the formula for triangular numbers?
The formula for the nth triangular number is Tn = n(n+1)/2. This formula is derived from the sum of an arithmetic series.
6. How many triangular numbers are there between 1 and 100?
To find this, we solve the inequality 1 ≤ n(n+1)/2 ≤ 100 for integer values of n. This gives us n values from 1 to 13, so there are 13 triangular numbers between 1 and 100.
7. What is the relationship between triangular numbers and square numbers?
Some numbers are both triangular and square (e.g., 1, 36, 1225...). There is an interesting mathematical relationship between the two sequences, though they are distinct. The formula to find numbers that are both triangular and square involves solving a Pell equation.
8. How are triangular numbers used in real-life situations?
Triangular numbers appear in various applications: bowling pins are arranged in a triangular formation, representing triangular numbers. They're also used in combinatorics problems (like calculating handshakes in a group), and in some algebraic and geometric concepts.
9. What is the sum of the first n triangular numbers?
The sum of the first n triangular numbers is given by the formula: n(n+1)(n+2)/6. This is also known as a tetrahedral number.
10. How do triangular numbers relate to Pascal's Triangle?
Triangular numbers are found in Pascal's Triangle. They correspond to the values in the third diagonal (excluding the initial 1).
11. Are there negative triangular numbers?
While the typical definition uses positive integers, the formula n(n+1)/2 can produce negative values if 'n' is a negative integer. However, these are usually not considered in elementary discussions of triangular numbers.
12. What is a geometric proof of the triangular number formula?
A visual proof involves arranging dots in a triangle. By mirroring the triangle and forming a rectangle, you can see that the total number of dots in the rectangle is n(n+1), and the triangle is half of that, resulting in the formula n(n+1)/2.
