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 N^{th} term of the index of smallest triangular number with N digits will be \lfloor \sqrt{2*10^{n-1} \rceil}

Below is the implementation of the above approach:

C++

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// 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;
}

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Java

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// 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 

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Python3

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# 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

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C#

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// 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

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Output:

14

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