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Finding n-th term of series 3, 13, 42, 108, 235…
  • Difficulty Level : Medium
  • Last Updated : 25 Mar, 2021

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 <bits/stdc++.h>
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




<?php
// PHP  program to find n-th term of
// series 3, 13, 42, 108, 235…
 
// Function to generate a fixed number
 
function magicOfSequence($N)
{
    $sum = 0;
    for ($i = 1; $i <= $N; $i++)
        $sum += ($i*$i*$i + $i*2);
    return $sum;
}
 
// Driver Method
 
    $N = 4;
    echo  magicOfSequence($N);
 
 
// This code is contributed by m_kit   
?>

Javascript




<script>
// JavaScript program to find n-th term of
// series 3, 13, 42, 108, 235…
 
// Function to generate a fixed number
function magicOfSequence( N)
{
    let sum = 0;
    for (let i = 1; i <= N; i++)
        sum += (i*i*i + i*2);
    return sum;   
}
 
// Driver Function
 
    let N = 4;
    document.write(magicOfSequence(N));
     
// This code contributed by Rajput-Ji
 
</script>
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 <iostream>
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




<?php
// A formula based PHP program
// to find sum of series with
// cubes of first n natural numbers
function magicOfSequence($N)
{
    return ($N * ($N + 1) / 2) + 2 * $N;
}
 
// Driver Code
$N = 6;
echo magicOfSequence($N) . "\n";
 
// This code is contributed by mits
?>

Javascript




<script>
// A formula based Java program to find sum
// of series with cubes of first n natural
// numbers
 
    function magicOfSequence( N)
    {
        return (N * (N + 1) / 2) + 2 * N;
    }
 
    // Driver Function    
    let N = 6;
    document.write(magicOfSequence(N));
 
// This code is contributed by 29AjayKumar 
</script>
Output: 
33

 

Time Complexity: O(1)
 

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