C++ Programe to find n-th term in series 1 2 2 3 3 3 4

In this problem, we are given an integer N. The task is to find n-th term in series 1 2 2 3 3 3 4….

Let’s take an example to understand the problem,

Input

N = 6

Output

3

Explanation

The series upto nth term is 1, 2, 2, 3, 3, 3, ...

Solution Approach

A simple approach to solve the problem is by using a nested loop. The outer for loop is from 1 to n. And the inner loop is from 1 to i (iterator of outer loop). For each iteration in the inner loop, count the number of elements of the series and return the value of i when count is equal to n.

A more efficient approach to solve the problem is using pattern positions. The elements of the sequence is with their positions in the series are −

Element 1: position 1
Element 2: position 2, 3
Element 3: position 4, 5, 6
Element 4: position 7, 8, 9, 10

For these values, we can create a series using the last position of element in the series which is,

1, 3, 6, 10, 15, 21, 28, ….

x appears in term, 1 + 2 + 3 + … + (x-2) + (x-1)...

This can be generalised as n = x*(x-1)/2

2n = x² - x => x² - x - 2n = 0

Solve the equation using formula for solution of quadratic equation,

$$x=1/2*(1+\sqrt{1+8*n)}$$

Program to illustrate the working of our solution,

Example

 Live Demo

#include <bits/stdc++.h>
using namespace std;
int findNthTerm(int n) {
   int x = (((1) + (double)sqrt(1 + (8 * n))) / 2);
   return x;
}
int main(){
   int n = 12;
   cout<<"The series is 1, 2, 2, 3, 3, 3, 4, 4, ...\n";
   cout<<n<<"th term of the series is "<<findNthTerm(n);
   return 0;
}

Output

The series is 1, 2, 2, 3, 3, 3, 4, 4, ...
12th term of the series is 5