# 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 (t_{n}) of the series is of the form,

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

So, t_{n} = A(n – 1)(n – 2)(n – 3) + B(n – 1)(n – 2) + C(n – 1) + D

Now,

t_{1} = D = 1

t_{2} = C (2 – 1) + D = 2

t_{3} = 2B + 2C + D = 6

t_{4} = 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 t_{n} 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<iostream> ` `#include<math.h> ` `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); ` `} ` |

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## 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 ` |

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## 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)) ` |

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## 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 ` |

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## PHP

`<?php ` `// PHP program to find Nth ` `// term of the series: ` `// 1 + 2 + 6 + 15 + 31 + 56 + .. ` ` ` `// calculate Nth term of given series ` `function` `Nth_Term(` `$n` `) ` `{ ` ` ` `return` `(2 * pow(` `$n` `, 3) - 3 * ` ` ` `pow(` `$n` `, 2) + ` `$n` `+ 6) / 6; ` `} ` ` ` `// Driver code ` `$N` `= 8; ` ` ` `echo` `Nth_Term(` `$N` `); ` ` ` `// This code is contributed by ` `// Shashank_Sharma ` `?> ` |

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**Output:**

141

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