Triangular Numbers

• Difficulty Level : Basic
• Last Updated : 26 Mar, 2021

A number is termed as triangular number if we can represent it in the form of triangular grid of points such that the points form an equilateral triangle and each row contains as many points as the row number, i.e., the first row has one point, second row has two points, third row has three points and so on. The starting triangular numbers are 1, 3 (1+2), 6 (1+2+3), 10 (1+2+3+4). Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

How to check if a number is Triangular?
The idea is based on the fact that n’th triangular number can be written as sum of n natural numbers, that is n*(n+1)/2. The reason for this is simple, base line of triangular grid has n dots, line above base has (n-1) dots and so on.

Method 1 (Simple)
We start with 1 and check if the number is equal to 1. If it is not, we add 2 to make it 3 and recheck with the number. We repeat this procedure until the sum remains less than or equal to the number that is to be checked for being triangular.
Following is the implementations to check if a number is triangular number.

C++

 // C++ program to check if a number is a triangular number // using simple approach. #include using namespace std;   // Returns true if 'num' is triangular, else false bool isTriangular(int num) {     // Base case     if (num < 0)         return false;       // A Triangular number must be sum of first n     // natural numbers     int sum = 0;     for (int n=1; sum<=num; n++)     {         sum = sum + n;         if (sum==num)             return true;     }       return false; }   // Driver code int main() {     int n = 55;     if (isTriangular(n))         cout << "The number is a triangular number";     else         cout << "The number is NOT a triangular number";       return 0; }

Java

 // Java program to check if a // number is a triangular number // using simple approach class GFG {           // Returns true if 'num' is     // triangular, else false     static boolean isTriangular(int num)     {         // Base case         if (num < 0)             return false;               // A Triangular number must be         // sum of first n natural numbers         int sum = 0;                   for (int n = 1; sum <= num; n++)         {             sum = sum + n;             if (sum == num)                 return true;         }               return false;     }           // Driver code     public static void main (String[] args)     {         int n = 55;         if (isTriangular(n))             System.out.print("The number "                 + "is a triangular number");         else             System.out.print("The number"              + " is NOT a triangular number");     } }   // This code is contributed // by Anant Agarwal.

Python3

 # Python3 program to check if a number is a # triangular number using simple approach.   # Returns True if 'num' is triangular, else False def isTriangular(num):       # Base case     if (num < 0):         return False       # A Triangular number must be     # sum of first n natural numbers     sum, n = 0, 1       while(sum <= num):               sum = sum + n         if (sum == num):             return True         n += 1       return False   # Driver code n = 55 if (isTriangular(n)):     print("The number is a triangular number") else:     print("The number is NOT a triangular number")   # This code is contributed by Smitha Dinesh Semwal.

C#

 // C# program to check if a number is a // triangular number using simple approach using System;   class GFG {           // Returns true if 'num' is     // triangular, else false     static bool isTriangular(int num)     {         // Base case         if (num < 0)             return false;               // A Triangular number must be         // sum of first n natural numbers         int sum = 0;                   for (int n = 1; sum <= num; n++)         {             sum = sum + n;             if (sum == num)                 return true;         }               return false;     }           // Driver code     public static void Main ()     {         int n = 55;                   if (isTriangular(n))             Console.WriteLine("The number "                 + "is a triangular number");         else             Console.WriteLine("The number"             + " is NOT a triangular number");     } }   // This code is contributed by vt_m.



Javascript



Output:

The number is a triangular number

Method 2 (Using Quadratic Equation Root Formula)
We form a quadratic equation by equating the number to the formula of sum of first ‘n’ natural numbers, and if we get atleast one value of ‘n’ that is a natural number, we say that the number is a triangular number.

Let the input number be 'num'. We consider,

n*(n+1) = num

as,

n2 + n + (-2 * num) = 0

Below is the implementation of above idea.

C++

 // C++ program to check if a number is a triangular number // using quadratic equation. #include using namespace std;   // Returns true if num is triangular bool isTriangular(int num) {     if (num < 0)         return false;       // Considering the equation n*(n+1)/2 = num     // The equation is  : a(n^2) + bn + c = 0";     int c = (-2 * num);     int b = 1, a = 1;     int d = (b * b) - (4 * a * c);       if (d < 0)         return false;       // Find roots of equation     float root1 = ( -b + sqrt(d)) / (2 * a);     float root2 = ( -b - sqrt(d)) / (2 * a);       // checking if root1 is natural     if (root1 > 0 && floor(root1) == root1)         return true;       // checking if root2 is natural     if (root2 > 0 && floor(root2) == root2)         return true;       return false; }   // Driver code int main() {     int num = 55;     if (isTriangular(num))         cout << "The number is a triangular number";     else         cout << "The number is NOT a triangular number";       return 0; }

Java

 // Java program to check if a number is a // triangular number using quadratic equation. import java.io.*;   class GFG {       // Returns true if num is triangular     static boolean isTriangular(int num)     {         if (num < 0)             return false;               // Considering the equation         // n*(n+1)/2 = num         // The equation is :         // a(n^2) + bn + c = 0";         int c = (-2 * num);         int b = 1, a = 1;         int d = (b * b) - (4 * a * c);               if (d < 0)             return false;               // Find roots of equation         float root1 = ( -b +            (float)Math.sqrt(d)) / (2 * a);                      float root2 = ( -b -            (float)Math.sqrt(d)) / (2 * a);               // checking if root1 is natural         if (root1 > 0 && Math.floor(root1)                                   == root1)             return true;               // checking if root2 is natural         if (root2 > 0 && Math.floor(root2)                                   == root2)             return true;               return false;     }           // Driver code     public static void main (String[] args) {         int num = 55;         if (isTriangular(num))             System.out.println("The number is"                     + " a triangular number");         else             System.out.println ("The number "               + "is NOT a triangular number");     } }   //This code is contributed by vt_m.

Python3

 # Python3 program to check if a number is a # triangular number using quadratic equation. import math   # Returns True if num is triangular def isTriangular(num):       if (num < 0):         return False       # Considering the equation n*(n+1)/2 = num     # The equation is : a(n^2) + bn + c = 0     c = (-2 * num)     b, a = 1, 1     d = (b * b) - (4 * a * c)       if (d < 0):         return False       # Find roots of equation     root1 = ( -b + math.sqrt(d)) / (2 * a)     root2 = ( -b - math.sqrt(d)) / (2 * a)       # checking if root1 is natural     if (root1 > 0 and math.floor(root1) == root1):         return True       # checking if root2 is natural     if (root2 > 0 and math.floor(root2) == root2):         return True       return False     # Driver code n = 55 if (isTriangular(n)):     print("The number is a triangular number") else:     print("The number is NOT a triangular number")   # This code is contributed by Smitha Dinesh Semwal

C#

 // C# program to check if a number is a triangular // number using quadratic equation. using System;   class GFG {           // Returns true if num is triangular     static bool isTriangular(int num)     {         if (num < 0)             return false;               // Considering the equation n*(n+1)/2 = num         // The equation is : a(n^2) + bn + c = 0";         int c = (-2 * num);         int b = 1, a = 1;         int d = (b * b) - (4 * a * c);               if (d < 0)             return false;               // Find roots of equation         float root1 = ( -b + (float)Math.Sqrt(d))                                         / (2 * a);                                                   float root2 = ( -b - (float)Math.Sqrt(d))                                          / (2 * a);               // checking if root1 is natural         if (root1 > 0 && Math.Floor(root1) == root1)             return true;               // checking if root2 is natural         if (root2 > 0 && Math.Floor(root2) == root2)             return true;               return false;     }           // Driver code     public static void Main () {                   int num = 55;         if (isTriangular(num))             Console.WriteLine("The number is a "                             + "triangular number");         else             Console.WriteLine ("The number is NOT "                           + "a triangular number");     } }   //This code is contributed by vt_m.

PHP

 0 && floor(\$root1) == \$root1)         return true;       // checking if root2 is natural     if (\$root2 > 0 && floor(\$root2) == \$root2)         return true;       return false; }   // Driver code \$num = 55; if (isTriangular(\$num))     echo("The number is" .          " a triangular number"); else     echo ("The number " .           "is NOT a triangular number");   // This code is contributed // by Code_Mech. ?>

Javascript



Output:

The number is a triangular number