Index of smallest triangular number with N digits

Given a number N, the task is to find the index of smallest triangular number with N digits.

A number is termed as 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, the second row has two points, the third row has three points and so on. The starting triangular numbers are 1, 3, 6, 10, 15, 21, 28………… 

Examples:

Input: N = 2 
Output: 4 
Smallest triangular number with 2 digits = 10, and 4 is the index of 10.

Input: N = 3 
Output: 14 
Smallest triangular number with 
3 digits = 105, and 14 is the index of 105. 

Approach: The key observation in the problem is that the index of smallest triangular numbers with N digits form a series which is – 

1, 4, 14, 45, 141...

The term of the index of smallest triangular number with N digits will be 
Below is the implementation of the above approach:

C++

// C++ implementation of
// the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to return index of smallest
// triangular no n digits
int findIndex(int n)
{
    float x = sqrt(2 * pow(10, (n - 1)));
    return round(x);
}

// Driver Code
int main()
{
    int n = 3;
    cout << findIndex(n);

    return 0;
}

Java

// Java implementation of the above approach
class GFG{

// Function to return index of smallest
// triangular no n digits
static double findIndex(int n)
{
    double x = Math.sqrt(2 * Math.pow(10, (n - 1)));
    return Math.round(x);
}

// Driver code
public static void main(String[] args)
{
    int n = 3;
    System.out.print(findIndex(n));
}
}

// This code is contributed by shubham

Python3

# Python3 implementation of
# the above approach
import math

# Function to return index of smallest
# triangular no n digits
def findIndex(n):

    x = math.sqrt(2 * math.pow(10, (n - 1)));
    return round(x);

# Driver Code
n = 3;
print(findIndex(n));

# This code is contributed by Code_Mech

C#

// C# implementation of the above approach
using System;
class GFG{

// Function to return index of smallest
// triangular no n digits
static double findIndex(int n)
{
    double x = Math.Sqrt(2 * Math.Pow(10, (n - 1)));
    return Math.Round(x);
}

// Driver code
public static void Main(String[] args)
{
    int n = 3;
    Console.Write(findIndex(n));
}
}

// This code is contributed by AbhiThakur

Javascript

<script>

// Javascript implementation of the above approach


    // Function to return index of smallest
    // triangular no n digits
    function findIndex( n) {
        let x = Math.sqrt(2 * Math.pow(10, (n - 1)));
        return Math.round(x);
    }

    // Driver code

        let n = 3;
        document.write(findIndex(n));

// This code is contributed by todaysgaurav

</script>

Output:

14

Time complexity: O(logn)
Auxiliary space: O(1)

Python program to find the index of the smallest triangular number with N digits without taking input:

Approach steps:

1.Import the math module.
2.Define a function smallest_triangular_index that takes an integer n as input. The function will return the index of the smallest triangular number with n digits.
3.Calculate the minimum triangular number with n digits by using the formula min_triangular = ceil(sqrt(8 * 10^(n-1) + 1) – 1) / 2, where ceil is the ceiling function, sqrt is the square root function, and ^ is the exponentiation operator. 

4.This formula is derived from the fact that the kth triangular number is equal to (k * (k + 1)) / 2, so the minimum triangular number with n digits will be greater than or equal to (10^(n-1) – 1) / 2. We can solve for k using the quadratic formula and take the ceiling of the positive root to find the minimum value of k that satisfies this inequality.
5.Return the value of min_triangular as the index of the smallest triangular number with n digits.
6.In the example usage, create an integer n and call the smallest_triangular_index function with this argument. Finally, print the index of the smallest triangular number with n digits.

Python3

# program to find the index of the smallest triangular number with N digits

import math

def smallest_triangular_index(n):
    # calculate the minimum triangular number with N digits
    min_triangular = int(math.ceil(math.sqrt(8 * math.pow(10, n-1) + 1) - 1) / 2)

    # return the index of the minimum triangular number
    return min_triangular

# example usage
n = 3
index = smallest_triangular_index(n)
print("Index of the smallest triangular number with", n, "digits is", index)

Output

Index of the smallest triangular number with 3 digits is 14

Time complexity: O(1).

Space complexity: O(1).