A number is termed a triangular number if we can represent it in the form of a 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).
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.
Examples:
Input: 10
Output: True
Explanation: 10 is a triangular number because it can be written as the sum of the first 4 natural numbers: 1 + 2 + 3 + 4 = 10.
Input: 8
Output: False
Explanation: 8 is not a triangular number because no sum of the first n natural numbers equals 8. The closest triangular numbers are 6 (1+2+3) and 10 (1+2+3+4).
[Naive Approach] Iterative Summation Method - O(n) time and O(1) space
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.
Below is the implementation of above approach
C++
// C++ program to check if a number is a triangular number
// using simple approach.
#include <iostream>
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 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 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# 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.
<script>
// javascript program to check if a number is a triangular number
// using simple approach.
// Returns true if 'num' is triangular, else false
function isTriangular(num)
{
// Base case
if (num < 0)
return false;
// A Triangular number must be sum of first n
// natural numbers
let sum = 0;
for (let n = 1; sum <= num; n++)
{
sum = sum + n;
if (sum == num)
return true;
}
return false;
}
// Driver code
let n = 55;
if (isTriangular(n))
document.write( "The number is a triangular number");
else
document.write( "The number is NOT a triangular number");
// This code is contributed by aashish1995
</script>
<?php
// PHP program to check if a number is a
// triangular number using simple approach.
// Returns true if 'num' is triangular,
// else false
function isTriangular( $num)
{
// Base case
if ($num < 0)
return false;
// A Triangular number must be
// sum of first n natural numbers
$sum = 0;
for ($n = 1; $sum <= $num; $n++)
{
$sum = $sum + $n;
if ($sum == $num)
return true;
}
return false;
}
// Driver code
$n = 55;
if (isTriangular($n))
echo "The number is a triangular number";
else
echo "The number is NOT a triangular number";
// This code is contributed by Rajput-Ji
?>
OutputThe number is a triangular number
[Expected Approach] Quadratic Equation Root Formula Method - O(log n) time and O(1) Space
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))/2 = num
as,
n2 + n + (-2 * num) = 0
Below is the implementation of above approach.
C++
// C++ program to check if a number is a triangular number
// using quadratic equation.
#include <bits/stdc++.h>
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 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 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# 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.
<script>
// javascript program to check if a number is a
// triangular number using quadratic equation.
// Returns true if num is triangular
function isTriangular(num)
{
if (num < 0)
return false;
// Considering the equation
// n*(n+1)/2 = num
// The equation is :
// a(n^2) + bn + c = 0";
var c = (-2 * num);
var b = 1, a = 1;
var d = (b * b) - (4 * a * c);
if (d < 0)
return false;
// Find roots of equation
var root1 = (-b + Math.sqrt(d)) / (2 * a);
var root2 = (-b - 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
var num = 55;
if (isTriangular(num))
document.write("The number is" + " a triangular number");
else
document.write("The number " + "is NOT a triangular number");
// This code is contributed by Rajput-Ji
</script>
<?php
// PHP program to check if a number is a
// triangular number using quadratic equation.
// Returns true if num is triangular
function 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 = 1; $a = 1;
$d = ($b * $b) - (4 * $a * $c);
if ($d < 0)
return false;
// Find roots of equation
$root1 = (-$b + (float)sqrt($d)) / (2 * $a);
$root2 = (-$b - (float)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
$num = 55;
if (isTriangular($num))
echo("The number is" .
" a triangular number");
else
echo ("The number " .
"is NOT a triangular number");
// This code is contributed
// by Code_Mech.
?>
OutputThe number is a triangular number
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