Program to Find the Nth Term of the Series 1, 3, 12, 60, 360…

Table of Contents#

1. Understanding the Series
2. Mathematical Formula for the Nth Term
3. Implementing the Program in Python
4. Implementing the Program in Java
5. Implementing the Program in C++
6. Best Practices and Common Pitfalls
7. Example Usage
8. Conclusion
9. References

1. Understanding the Series#

Let's take a closer look at the given series: 1, 3, 12, 60, 360…

• The first term (a_1 = 1)
• The second term (a_2=3 = 1\times3)
• The third term (a_3 = 12=3\times4)
• The fourth term (a_4 = 60 = 12\times5)
• The fifth term (a_5=360 = 60\times6)

We can observe that each term in the series is obtained by multiplying the previous term by an increasing integer starting from 3.

2. Mathematical Formula for the Nth Term#

Based on the pattern we observed, we can derive a formula for the Nth term of the series.

Let (a_n) be the Nth term of the series.

• For (n = 1), (a_1=1)
• For (n > 1), (a_n=a_{n - 1}\times(n + 1))

We can also write the formula in a more explicit way by expanding the recurrence relation:

(a_n=1\times3\times4\times\cdots\times(n + 1)=\frac{(n + 1)!}{2})

3. Implementing the Program in Python#

def nth_term(n):
    if n == 1:
        return 1
    term = 1
    for i in range(3, n + 2):
        term *= i
    return term
 
# Example usage
n = 5
print(f"The {n}th term of the series is {nth_term(n)}")

In this Python code, we first check if (n) is 1. If it is, we return 1. Otherwise, we use a for loop to calculate the product of numbers from 3 to (n + 1).

4. Implementing the Program in Java#

public class SeriesNthTerm {
    public static int nthTerm(int n) {
        if (n == 1) {
            return 1;
        }
        int term = 1;
        for (int i = 3; i <= n + 1; i++) {
            term *= i;
        }
        return term;
    }
 
    public static void main(String[] args) {
        int n = 5;
        System.out.printf("The %dth term of the series is %d%n", n, nthTerm(n));
    }
}

In the Java code, we have a method nthTerm that calculates the Nth term of the series. In the main method, we call this method and print the result.

5. Implementing the Program in C++#

#include <iostream>
using namespace std;
 
int nthTerm(int n) {
    if (n == 1) {
        return 1;
    }
    int term = 1;
    for (int i = 3; i <= n + 1; i++) {
        term *= i;
    }
    return term;
}
 
int main() {
    int n = 5;
    cout << "The " << n << "th term of the series is " << nthTerm(n) << endl;
    return 0;
}

The C++ code is similar to the Java code. We have a function nthTerm to calculate the Nth term and print the result in the main function.

6. Best Practices and Common Pitfalls#

Best Practices#

• Error Handling: Always check for invalid input. For example, if (n) is less than 1, the program should handle it gracefully.
• Code Readability: Use meaningful variable names and add comments to make the code easy to understand.
• Efficiency: The current implementation has a time complexity of (O(n)). If you need to calculate multiple terms frequently, you can use memoization to avoid redundant calculations.

Common Pitfalls#

• Integer Overflow: If (n) is large, the result may exceed the maximum value that can be stored in an integer data type. You can use a larger data type like long long in C++ or BigInteger in Java to avoid this issue.

7. Example Usage#

Let's say we want to find the 6th term of the series.

• Python:

n = 6
print(f"The {n}th term of the series is {nth_term(n)}")

• Java:

int n = 6;
System.out.printf("The %dth term of the series is %d%n", n, nthTerm(n));

• C++:

int n = 6;
cout << "The " << n << "th term of the series is " << nthTerm(n) << endl;

8. Conclusion#

In this blog, we have learned how to find the Nth term of the series 1, 3, 12, 60, 360… We first analyzed the pattern of the series, derived a mathematical formula, and then implemented the program in Python, Java, and C++. We also discussed best practices and common pitfalls to keep in mind when writing such programs.

9. References#

• Python Documentation: https://docs.python.org/3/
• Java Documentation: https://docs.oracle.com/javase/8/docs/
• C++ Documentation: https://en.cppreference.com/w/