Introduction to the Sum of the First 50 Odd Natural Numbers

The sum of the first 50 odd natural numbers is a classical problem in mathematics that can be computed using different methods. This article will explore how to calculate this sum using various approaches, including the formula for the sum of an arithmetic series and a direct computational method. Understanding these methods can help educators and students alike in both classroom settings and practical applications.

Formula for Sum of Odd Natural Numbers

A common method to find the sum of the first n odd natural numbers is by utilizing the formula:

Sum n2

For the first 50 odd natural numbers, setting n 50, the calculation becomes:

Sum 502 2500

Calculation Using Arithmetic Series Formula

The sum of an arithmetic series can also be calculated by the formula:

S n/2 × [a1 an]

For odd natural numbers, the common difference d is 2, and the first term a1 is 1. For the first 50 odd natural numbers, n is 50. Substituting these values into the formula, we get:

S 50/2 × [1 99] 25 × 100 2500

Here, 99 is the last term of the first 50 odd natural numbers, which can be calculated as 2 × 50 - 1.

Pattern of Summation

A more intuitive approach is to observe a pattern: the sum of the first n odd natural numbers equals n2. This can be demonstrated through examples:

The sum of the first 2 odd numbers: 1 3 4 22 The sum of the first 3 odd numbers: 1 3 5 9 32 The sum of the first 4 odd numbers: 1 3 5 7 16 42

Following this pattern, the sum of the first 50 odd numbers is:

1 3 5 ... 97 99 502 2500

Finding the Last Term

The last term in the sequence of the first 50 odd natural numbers is 99, which can be determined using the formula for the n-th term of an arithmetic sequence:

an a1 (n - 1) × d

Substituting a1 1, d 2, and n 50, we get:

a50 1 (50 - 1) × 2 1 49 × 2 1 98 99

Computational Verification

To further verify the sum, a simple C program can be written to calculate the sum:

```c #include stdio.h int main() { tint n 50, sum 0, a 1; tfor (int i 1; i

The output of this program will be:

The sum is: 2500

Conclusion and FAQs

In conclusion, the sum of the first 50 odd natural numbers is 2500, which can be verified using multiple methods including the arithmetic series formula, mathematical pattern recognition, and a simple computational approach. Understanding these methods is essential for solving similar mathematical problems and can be beneficial in various fields such as data science, engineering, and education.

Frequently Asked Questions (FAQs)

Q: What is the formula for the sum of the first n odd natural numbers?
A: The formula is Sum n2. Q: How do you calculate the sum using an arithmetic series?
A: You can use the formula S n/2 × [a1 an], where n is the number of terms, a1 is the first term, and an is the last term. Q: What is the last term in the sequence of the first 50 odd natural numbers?
A: The last term is 99, which can be calculated using the formula an a1 (n - 1) × d.