Un número se denomina número triangular si podemos representarlo en forma de rejilla triangular de puntos tal que los puntos formen un triángulo equilátero y cada fila contenga tantos puntos como el número de la fila, es decir, la primera fila tiene un punto, la segunda la fila tiene dos puntos, la tercera fila tiene tres puntos y así sucesivamente. Los números triangulares iniciales son 1, 3 (1+2), 6 (1+2+3), 10 (1+2+3+4).
¿Cómo comprobar si un número es triangular?
La idea se basa en el hecho de que el n-ésimo número triangular se puede escribir como la suma de n números naturales, es decir, n*(n+1)/2. La razón de esto es simple, la línea base de la cuadrícula triangular tiene n puntos, la línea sobre la base tiene (n-1) puntos y así sucesivamente.
Método 1 (Simple)
Comenzamos con 1 y verificamos si el número es igual a 1. Si no lo es, sumamos 2 para convertirlo en 3 y volvemos a verificar con el número. Repetimos este procedimiento hasta que la suma quede menor o igual que el número que se quiere comprobar que es triangular.
A continuación se muestran las implementaciones para verificar si un número es un número triangular.
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
// 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.
PHP
<?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
?>
Javascript
<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>
Producción:
The number is a triangular number
Complejidad de tiempo: O(n)
Espacio auxiliar: O(1)
Método 2 (Uso de la fórmula raíz de la ecuación cuadrática)
Formamos una ecuación cuadrática igualando el número a la fórmula de la suma de los primeros ‘n’ números naturales, y si obtenemos al menos un valor de ‘n’ que es un número natural, decimos que el número es un número triangular.
Let the input number be 'num'. We consider, n*(n+1) = num as, n2 + n + (-2 * num) = 0
A continuación se muestra la implementación de la idea anterior.
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
// 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
<?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.
?>
Javascript
<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>
Producción:
The number is a triangular number
Complejidad de tiempo: O (logn)
Espacio auxiliar: O(1)
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Publicación traducida automáticamente
Artículo escrito por GeeksforGeeks-1 y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA