Sum of the series 1.2.3 + 2.3.4 + ... + n(n+1)(n+2)

Find the sum up to n terms of the series: 1.2.3 + 2.3.4 + ... + n(n+1)(n+2). In this 1.2.3 represent the first term and 2.3.4 represent the second term .

Examples : 

Input : 2
Output : 30
Explanation: 1.2.3 + 2.3.4 = 6 + 24 = 30

Input : 3
Output : 90

Simple Approach We run a loop for i = 1 to n, and find the sum of (i)*(i+1)*(i+2). 
And at the end display the sum .
 

// CPP program to find sum of the series
// 1.2.3 + 2.3.4 + 3.4.5 + ...
#include <bits/stdc++.h>
using namespace std;

int sumofseries(int n)
{
    int res = 0;
    for (int i = 1; i <= n; i++) 
        res += (i) * (i + 1) * (i + 2);    
    return res;
}

// Driver Code
int main()
{
    cout << sumofseries(3) << endl;
    return 0;
}

// Java program to find sum of the series
// 1.2.3 + 2.3.4 + 3.4.5 + ...
import java.io.*;
import java.math.*;

class GFG 
{

    static int sumofseries(int n)
    {
    int res = 0;
    for (int i = 1; i <= n; i++) 
        res += (i) * (i + 1) * (i + 2); 
    return res;
    }

    // Driver Code
    public static void main(String[] args)
    {
        System.out.println(sumofseries(3));
    }
}

# Python 3 program to find sum of the series
# 1.2.3 + 2.3.4 + 3.4.5 + ...

def sumofseries(n):

    res = 0
    for i in range(1, n+1): 
        res += (i) * (i + 1) * (i + 2) 
    return res

# Driver Program
print(sumofseries(3)) 

# This code is contributed
# by Smitha Dinesh Semwal

// Java program to find sum of the series
// 1.2.3 + 2.3.4 + 3.4.5 + ...
using System;

class GFG 
{

    static int sumofseries(int n)
    {
        int res = 0;
        for (int i = 1; i <= n; i++) 
            res += (i) * (i + 1) * (i + 2); 
        return res;
    }

    // Driver Code
    public static void Main()
    {
        Console.WriteLine(sumofseries(3));
    }
}

// This code is contributed by vt_m.

<?php
// PHP program to find
// sum of the series
// 1.2.3 + 2.3.4 + 3.4.5 + ...

function sumofseries($n)
{
    $res = 0;
    for ($i = 1; $i <= $n; $i++) 
        $res += ($i) * ($i + 1) * 
                       ($i + 2); 
    return $res;
}

    // Driver Code
    echo sumofseries(3);

//This code is contributed by anuj_67.
?>

<script>
// JavaScript program to find sum of the series
// 1.2.3 + 2.3.4 + 3.4.5 + ...

    function sumofseries(n)
    {
    let res = 0;
    for (let i = 1; i <= n; i++) 
        res += (i) * (i + 1) * (i + 2); 
    return res;
    } 

// Driver Code

    document.write(sumofseries(3));

// This code is contributed by code_hunt.
</script>

90

Time Complexity: O(N)
Auxiliary Space: O(1)

Efficient Approach

Using Efficient Approach we know that we have to find = summation of( (n)*(n+1)*(n+2) )
 

S_n = summation[ (n)*(n+1)*(n+2) ] 
S_n = summation [n³ + 2*n² + n² + 2*n]
We know sum of cubes of natural numbers is (n*(n+1))/2)², sum of squares of natural numbers is n * (n + 1) * (2n + 1) / 6 and sum of first n natural numbers is n(n+1)/2
S_n = ((n*(n+1))/2)² + 3((n)*(n+1)*(2*n+1)/6) + 2*((n)*(n+1)/2) 
So by evaluating the above we get, 
S_n = (n*(n+1)*(n+2)*(n+3)/4) 
Hence it has a O(1) complexity.

// Efficient CPP program to 
// find sum of the series 
// 1.2.3 + 2.3.4 + 3.4.5 + ...
#include <bits/stdc++.h>
using namespace std;

// function to calculate
// sum of series
int sumofseries(int n)
{
    return (n * (n + 1) * 
           (n + 2) * (n + 3) / 4);
}

// Driver Code
int main()
{
    cout << sumofseries(3) << endl;
    return 0;
}

// Efficient Java program to
// find sum of the series 
// 1.2.3 + 2.3.4 + 3.4.5 + ..
import java.io.*;
import java.math.*;

class GFG 
{
    static int sumofseries(int n)
    {
    return (n * (n + 1) * 
           (n + 2) * (n + 3) / 4);
    }

    // Driver Code
    public static void main(String[] args)
    {
        System.out.println(sumofseries(3));
    } 
}

# Efficient CPP program to find sum of the
# series 1.2.3 + 2.3.4 + 3.4.5 + ...

# function to calculate sum of series
def sumofseries(n):

    return int(n * (n + 1) * (n + 2) * (n + 3) / 4)


# Driver program
print(sumofseries(3))
    

# This code is contributed
# by Smitha Dinesh Semwal

// Efficient C# program to 
// find sum of the series 
// 1.2.3 + 2.3.4 + 3.4.5 + ..
using System;

class GFG 
{
    static int sumofseries(int n)
    {
    return (n * (n + 1) * 
           (n + 2) * (n + 3) / 4);
    }

    // Driver Code
    public static void Main()
    {
        Console.WriteLine(sumofseries(3));
    } 
}

// This code is contributed by anuj_67.

<?php
// Efficient CPP program
// to find sum of the
// series 1.2.3 + 2.3.4 
// + 3.4.5 + ...

// function to calculate
// sum of series
function sumofseries($n)
{
    return ($n * ($n + 1) * 
           ($n + 2) * ($n + 3) / 4);
}

    // Driver Code
    echo sumofseries(3);

// This code is contributed by anuj_67.
?>

<script>
// Efficient Javascript program
// to find sum of the
// series 1.2.3 + 2.3.4
// + 3.4.5 + ...

// function to calculate
// sum of series
function sumofseries(n)
{
    return (n * (n + 1) *
        (n + 2) * (n + 3) / 4);
}

    // Driver Code
    document.write(sumofseries(3));

// This code is contributed by gfgking

</script>

90

Time Complexity: O(1)
 Auxiliary Space: O(1)