# Finding n-th term of series 3, 13, 42, 108, 235…

Given a number n, find the n-th term in the series 3, 13, 42, 108, 235…

Examples:

Input : 3
Output : 42

Input : 4
Output : 108

Constraints:
1 <= T <= 100
1 <= N <= 100

Naive Approach :
The series basically represents sums of natural numbers cube and number of terms multiplied by 2. The first term is the sum of the single number. The second term is the sum of two numbers, and so on.

Examples:

n = 2
2nd term equals to sum of 1st term and 8 i.e
A2 = A1 + 23 + n*2
= 1 + 8 + 4
= 13

Similarly,
A3 = A2 + 33 + n*2
= 9 + 27 + 6
= 42 and so on..

A simple solution is to add the first n natural numbers cube and number of terms multiplied by 2.

## C++

 // C++ program to find n-th term of  // series 3, 13, 42, 108, 235… #include using namespace std;    // Function to generate a fixed number int magicOfSequence(int N) {     int sum = 0;     for (int i = 1; i <= N; i++)          sum += (i*i*i + i*2);     return sum;     }    // Driver Method int main() {     int N = 4;     cout << magicOfSequence(N) << endl;     return 0; }

## Java

 // Java Program to Finding n-th term // of series 3, 13, 42, 108, 235 ...    class GFG {    // Function to generate // a fixed number public static int magicOfSequence(int N) {     int sum = 0;     for (int i = 1; i <= N; i++)          sum += (i * i * i + i * 2);     return sum;  }    // Driver Method public static void main(String args[]) {     int N = 4;     System.out.println(magicOfSequence(N)); } }    // This code is contributed by Jaideep Pyne

## Python3

 # Python3 program to # find n-th term of  # series 3, 13, 42, 108, 235…    # Function to generate # a fixed number def magicOfSequence(N) :        sum = 0     for i in range(1, N + 1) :         sum += (i * i * i + i * 2)     return sum;     # Driver Code N = 4 print(magicOfSequence(N))    # This code is contributed by vij.

## C#

 // C# Program to Finding  // n-th term of series  // 3, 13, 42, 108, 235 ... using System;    class GFG {        // Function to generate // a fixed number public static int magicOfSequence(int N) {     int sum = 0;     for (int i = 1; i <= N; i++)          sum += (i * i * i + i * 2);     return sum;  }    // Driver Code static public void Main () {     int N = 4;     Console.WriteLine(magicOfSequence(N)); } }    // This code is contributed // by ajit

## PHP



Output:

120

Time Complexity of this solution is O(n).

Efficient approach :
We know sum of cubes of first n natural numbers is (n*(n+1)/2)2. We also know that if we multiply i-th term by 2 and add all, we get sum of n terms as 2*n.

So our result is (n*(n+1)/2)2 + 2*n.

Example :

For n = 4 sum by the formula is
(4 * (4 + 1 ) / 2)) ^ 2 + 2*4
= (4 * 5 / 2) ^ 2 + 8
= (10) ^ 2 + 8
= 100 + 8
= 108

For n = 6, sum by the formula is
(6 * (6 + 1 ) / 2)) ^ 2 + 2*6
= (6 * 7 / 2) ^ 2 + 12
= (21) ^ 2 + 12
= 441 + 12
= 453

## C++

 // A formula based C++ program to find sum // of series with cubes of first n natural // numbers #include using namespace std;    int magicOfSequence(int N) {     return (N * (N + 1) / 2) + 2 * N; }    // Driver Function int main() {     int N = 6;     cout << magicOfSequence(N);     return 0; }

## Java

 // A formula based Java program to find sum // of series with cubes of first n natural // numbers class GFG {            static int magicOfSequence(int N)     {         return (N * (N + 1) / 2) + 2 * N;     }            // Driver Function     public static void main(String[] args)     {         int N = 6;         System.out.println(magicOfSequence(N));     } }    // This code is contributed by Smitha.

## Python 3

 # A formula based Python program to find sum # of series with cubes of first n natural # numbers def magicOfSequence(N):        return (N * (N + 1) / 2) + 2 * N    # Driver Function N = 6 print(int(magicOfSequence(N)))    # This code is contributed by Smitha.

## C#

 // A formula based C# program to find sum // of series with cubes of first n natural // numbers using System;    class GFG {            static int magicOfSequence(int N)     {         return (N * (N + 1) / 2) + 2 * N;     }            // Driver Function     public static void Main()     {         int N = 6;         Console.Write(magicOfSequence(N));     } }    // This code is contributed by Smitha.

## PHP



Output:

33

Time Complexity: O(1)

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