# Find the Nth term of the series 1 + 2 + 6 + 15 + 31 + 56 + …

Given an integer . The task is to write a program to find the N-th term of the given series:

1 + 2 + 6 + 15 + 31 + 56 + ...

Examples:

Input : N = 8
Output : 141

Input : N = 20
Output : 2471


Approach:
The given series is:

1, 2, 6, 15, 31, 56, 92, 141, ...


First consecutive difference:

1 4 9 16 25 36 49 .......


Second consecutive difference:

3 5 7 9 11 13......


As the second consecutive difference is in AP, the nth term (tn) of the series is of the form,

A(n – 1)(n – 2)(n – 3) + B(n – 1)(n – 2) + C(n – 1) + D

So, tn = A(n – 1)(n – 2)(n – 3) + B(n – 1)(n – 2) + C(n – 1) + D

Now,
t1 = D = 1

t2 = C (2 – 1) + D = 2

t3 = 2B + 2C + D = 6

t4 = CA + 6B + 3C + D = 15

On solving the above four equations we get => A = 1/3, B = 3/2, C = 1, D = 1. On substituting these values tn and after simplifying we get, Below is the implementation of above approach:

## C++

 // C++ program to find Nth   // term of the series:  // 1 + 2 + 6 + 15 + 31 + 56 + ...  #include  #include  using namespace std;     // calculate Nth term of given series  int Nth_Term(int n)  {      return (2 * pow(n, 3) - 3 *                   pow(n, 2) + n + 6) / 6;  }     // Driver code  int main()  {      int N = 8;      cout << Nth_Term(N);  }

## Java

 // Java program to find Nth   // term of the series:   // 1 + 2 + 6 + 15 + 31 + 56 + ...   import java.util.*;  import java.lang.*;     class GFG  {  // calculate Nth term of given series   static double Nth_Term(int n)   {       return (2 * Math.pow(n, 3) - 3 *                   Math.pow(n, 2) + n + 6) / 6;   }      // Driver code   static public void main (String args[])  {      int N = 8;       System.out.println(Nth_Term(N));   }  }     // This code is contributed  // by Akanksha Rai

## Python3

 # Python program to find Nth term of the series:  # 1 + 2 + 6 + 15 + 31 + 56 + ...     # calculate Nth term of given series  def Nth_Term(n):      return (2 * pow(n, 3) - 3 * pow(n, 2) + n + 6) // 6    # Driver code  N = 8 print(Nth_Term(N))

## C#

 // C# program to find Nth   // term of the series:   // 1 + 2 + 6 + 15 + 31 + 56 + ...   using System;     class GFG  {  // calculate Nth term of given series   static double Nth_Term(int n)   {       return (2 * Math.Pow(n, 3) - 3 *                   Math.Pow(n, 2) + n + 6) / 6;   }      // Driver code   static public void Main ()  {      int N = 8;       Console.WriteLine(Nth_Term(N));   }  }     // This code is contributed  // by Sach_Code

## PHP

 

Output:

141


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