Formula for the Sum of the First n Odd Numbers

Introduction

In mathematics, we often encounter series of numbers that follow specific patterns. One such series is the sequence of odd numbers: 1, 3, 5, 7, 9, …. The sum of the first n odd numbers is a useful concept with applications in various fields.

Derivation of the Formula

The formula for the sum of the first n odd numbers is given by:

“`
S_n = n^2
“`

where:

* S_n is the sum of the first n odd numbers
* n is the number of odd numbers to be summed

This formula can be derived using mathematical induction:

* **Base Case:** When n = 1, the sum of the first odd number is simply 1^2 = 1, which is true.
* **Inductive Step:** Assume that the formula is true for some integer k, i.e., S_k = k^2. Then, the sum of the first (k+1) odd numbers can be expressed as:

“`
S_(k+1) = S_k + (2k+1)
“`

Substituting S_k = k^2, we get:

“`
S_(k+1) = k^2 + (2k+1)
“`

Expanding and simplifying, we obtain:

“`
S_(k+1) = k^2 + 2k + 1
“`

“`
S_(k+1) = (k+1)^2
“`

This proves that if the formula is true for some integer k, it is also true for k+1. Therefore, by the principle of mathematical induction, the formula S_n = n^2 is valid for all positive integers n.

Applications

The formula for the sum of the first n odd numbers has numerous applications in fields such as:

* **Probability:** Calculating probabilities in certain games and scenarios
* **Physics:** Determining the center of mass of a system of odd numbers
* **Computer Science:** Solving algorithmic problems involving odd numbers

Examples

Let’s consider some examples to illustrate the formula:

* **Example 1:** Find the sum of the first 5 odd numbers.

Using the formula, we have:

“`
S_5 = 5^2 = 25
“`

Therefore, the sum of the first 5 odd numbers is 25.

* **Example 2:** If the sum of the first n odd numbers is 121, find the value of n.

Setting S_n = 121 in the formula, we get:

“`
121 = n^2
“`

Taking the square root of both sides, we find:

“`
n = 11
“`

Therefore, the sum of the first 11 odd numbers is 121.

Conclusion

The formula for the sum of the first n odd numbers (S_n = n^2) is a useful and versatile mathematical concept. It can be applied in various fields, including probability, physics, and computer science. By understanding the derivation and applications of this formula, we can effectively solve problems that involve series of odd numbers.

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