Program to find sum of series 1*2*3 + 2*3*4+ 3*4*5 + . . . + n*(n+1)*(n+2) in C++

In this problem, we are given a number n that defines the nth term of the series. Our task is to create a program to find sum of series 1*2*3 + 2*3*4+ 3*4*5 + . . . + n*(n+1)*(n+2) in C++.

Problem description − Here, we will find the sum till n term of the given series with is 1*2*3 + 2*3*4+ 3*4*5 + . . . + n*(n+1)*(n+2). This can be decoded as the summation of n*(n+1)*(n+2).

Let’s take an example to understand the problem

Input

n = 5

Output

420

Explanation

1*2*3 + 2*3*4 + 3*4*5 + 4*5*6 + 5*6*7 = 6 + 24 + 60 + 120 + 210 = 420

Solution Approach

A simple way to solve the problem is using a loop from 1 to n and find the product at each iteration and add it to the sumVar. After the loop ends, return the sumVar.

Algorithm

• Step 1 − loop i = 1 to n.
  • Step 1.1 − update sumVar, sumVar += i*(i+1)*(i+2)
• Step 2 − print sumVar.

Program to illustrate the working of our solution

Example

#include <iostream>
using namespace std;
int calcSeriesSum(int n){
   int sumVar = 0;
   for(int i = 1; i <= n; i++)
      sumVar = sumVar + ( (i)*(i+1)*(i+2) );
   return sumVar;
}
int main(){
   int n = 7;
   cout<<"The sum of series till "<<n<<" is "<<calcSeriesSum(n);
   return 0;
}

Output

The sum of series till 7 is 1260

This approach is not efficient as it takes time complexity of the order of N.

Another approach is using the mathematical formula for the sum of series. As we have discussed in the problem description, the series can be said the summation of (n)*(n+1)*(n+2).

Using this information let’s create a general formula for the sum.

$Sum =\sum_{\square=1}^\square\blacksquare((\square)\ast(\square+1)\ast(\square+2))$

$=\sum\lbrace{(n^2+n)(n+2)}\rbrace$

$=\sum\lbrace{n^3 + n^2 + 2n^2 + 2n}\rbrace$

$=\sum\lbrace{n^3 +3n^2 + 2n}\rbrace$

$=\sum_{\square=1}^\square\blacksquare\square^3+3\sum_{\square=1}^\square\blacksquare\square^2+2\sum_{\square=1}^\square\blacksquare\square^\blacksquare$

Now, using the general formula for the sums,

$\sum_{\square=1}^\square\blacksquare\square^3=\frac{(\square\ast(\square+1))^2}{2}$

$\sum_{\square=1}^\square\blacksquare\square^2=\frac{(\square\ast(\square+1)\ast(2\square+1))^\blacksquare}{6}$

$\sum_{\square=1}^\square\blacksquare\square^\blacksquare=\frac{(\square\ast(\square+1)^\blacksquare}{2}$

Adding all these to sum formula makes,

$Sum=\frac{(\square\ast(\square+1))^2}{2^2}+\frac{3(\square\ast(\square+1)\ast(2\square+1))^\blacksquare}{6}+\frac{2(\square\ast(\square+1))^\blacksquare}{2}$

$=\frac{(\square\ast(\square+1))^\blacksquare}{2}[( (n * (n+1))/2 ) + (3(2n+1)/3) + 2]$

$=\frac{(\square\ast(\square+1))^\blacksquare}{4}[n^2+ n + 4n + 2 + 4]$

$=\frac{(\square\ast(\square+1))^\blacksquare}{4}[n^2+ 5n + 6]$

$=\frac{(\square\ast(\square+1))^\blacksquare}{4}[(n+2)(n+3)]$

=¼[ (n)*(n+1)*(n+2)*(n+3) ]

The of the series till nth term is calculated using the formula,

¼[ (n)*(n+1)*(n+2)*(n+3) ]

Program to illustrate the working of our solution

Example

#include <iostream>
using namespace std;
int calcSeriesSum(int n){
   int sumVar = 0;
   sumVar = ( (n)*(n + 1)*(n + 2)*(n + 3)/4 );
   return sumVar;
}
int main(){
   int n = 7;
   cout<<"The sum of series till "<<n<<" is "<<calcSeriesSum(n);
   return 0;
}

Output

The sum of series till 7 is 1260