Sum of first N terms of Quadratic Sequence 3 + 7 + 13 + ... - GeeksforGeeks

08 Apr, 2021
Given a quadratic series as given below, the task is to find sum of first n terms of this series.
Sn = 3 + 7 + 13 + 21 + 31 + ..... + upto n terms
Examples: 
Input: N = 3
Output: 23

Input: N = 4
Output: 44
Approach: Let the series be represented as  
Sn = 3 + 7 + 13 + ... + tn
where  
Sn represents the sum of the series till n terms.
tn represents the nth term of the series.
Now, to formulate the series, the elements need to be formed by taking the difference of the consecutive elements of the series.
Equation 1: Sn = 3 + 7 + 13 + 21 + 31 +.....+ tn-1 + tn Equation 2: Sn = 0 + 3 + 7 + 13 + 21 + 31 + ...... + tn-1 + tn (writing the above series by shifting all elements to right by 1 position)  
Now, Subtract Equation 2 from Equation 1 i.e. (Equation 1 – Equation 2)  
Sn – Sn = (3 – 0) + (7 – 3) + (13 – 7) + (31 – 21) + ...... + (tn- tn-1) – tn => 0 = 3 + 4 + 6 + 8 + 10 + ...... + (tn – tn-1) – tn  
In above series, leaving 3 aside, terms starting from 4 to (tn – tn-1) will form an A.P.Since formula of sum of n terms of A.P. is:
Sn = n*(2*a + (n – 1)*d)/2
which implies, 
In series: 4 + 6 + 8 + ... + (tn – tn-1) AP is formed with (n-1) terms.  
Hence,  
Sum of this series: (n-1)*(2*4 + (n-2)*2)/2  
Therefore, the original series: 0 = 3 + (n-1)*(2*4 + (n-2)*2)/2 – tn where tn = n^2 + n + 1 which is the nth term.Therefore, 
Sum of first n terms of series will be:tn = n^2 + n + 1 Sn = (n^2) + n + (1) Sn = n*(n+1)*(n+2)/6 + n*(n+1)/2 + n Sn = n*(n^2 + 3*n + 5)/3  
Below is the implementation of above approach:  
C++
Java
Python3
C#
PHP
Javascript
// C++ program to find sum of first n terms #include <bits/stdc++.h>using namespace std; int calculateSum(int n){    // Sum = n*(n^2 + 3*n + 5)/3    return n * (pow(n, 2) + 3 * n + 5) / 3;} int main(){    // number of terms to be included in the sum    int n = 3;     // find the Sum    cout << "Sum = " << calculateSum(n);     return 0;}
// Java program to find sum of first n termsimport java.util.*; class solution{//function to calculate sum of n terms of the seriesstatic int calculateSum(int n){    // Sum = n*(n^2 + 3*n + 5)/3    return n * (int)  (Math.pow(n, 2) + 3 * n + 5 )/ 3;} public static void main(String arr[]){    // number of terms to be included in the sum    int n = 3;     // find the Sum    System.out.println("Sum = " +calculateSum(n)); }}
# Python 3 program to find sum# of first n termsfrom math import pow def calculateSum(n):         # Sum = n*(n^2 + 3*n + 5)/3    return n * (pow(n, 2) + 3 * n + 5) / 3 if __name__ == '__main__':         # number of terms to be included    # in the sum    n = 3     # find the Sum    print("Sum =", int(calculateSum(n))) # This code is contributed by# Sanjit_Prasad
// C# program to find sum of first n termsusing System;class gfg{ public double calculateSum(int n) {    // Sum = n*(n^2 + 3*n + 5)/3    return (n * (Math.Pow(n, 2) + 3 * n + 5) / 3);  }} //driver codeclass geek{ public static int Main() {     gfg g = new gfg();    // number of terms to be included in the sum    int n = 3;    //find the Sum    Console.WriteLine( "Sum = {0}", g.calculateSum(n));    return 0; }}
<?php// PHP program to find sum// of first n terms function calculateSum($n){    // Sum = n*(n^2 + 3*n + 5)/3    return $n * (pow($n, 2) + 3 *                     $n + 5) / 3;} // Driver Code // number of terms to be// included in the sum$n = 3; // find the Sumecho "Sum = " . calculateSum($n); // This code is contributed by mits?>
<script> // Javascript program to find sum of first n terms // Function to find the quadratic// equation whose roots are a and bfunction calculateSum(n){         // Sum = n*(n^2 + 3*n + 5)/3    return n * (Math.pow(n, 2) + 3 * n + 5 ) / 3;} // Driver Code // Number of terms to be// included in the sumvar n = 3; // Find the Sum document.write("Sum = " + calculateSum(n)); // This code is contributed by Ankita saini     </script>
Sum = 23
 
SURENDRA_GANGWAR
SoumikMondal
Mithun Kumar
Sanjit_Prasad
Akanksha_Rai
ankita_saini
series
series-sum
Mathematical
School Programming
Mathematical
series
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Merge two sorted arrays
Modulo Operator (%) in C/C++ with Examples
Prime Numbers
Sieve of Eratosthenes
Program to find GCD or HCF of two numbers
Python Dictionary
Arrays in C/C++
Inheritance in C++
Reverse a string in Java
C++ Classes and Objects