Proving the Relationship Between the Sum of Natural Numbers and the Sum of Odd Natural Numbers

In this article, we will delve into the mathematical relationship between the sum of the first n natural numbers and the sum of the first n odd natural numbers. Specifically, we aim to prove that the sum of the first n natural numbers is equal to n times the sum of the first n odd natural numbers divided by n. This relationship can be a valuable tool in various mathematical and computational applications.

The Sum of the First n Natural Numbers

Let us begin by defining the sum of the first n natural numbers. The sum of the first n natural numbers (1, 2, 3, ..., n) is given by the well-known formula:

Sn {1}{2}(1 n)n

The Sum of the First n Odd Natural Numbers

The sum of the first n odd natural numbers (1, 3, 5, ..., 2n-1) can be expressed using a different formula. This sum is given by:

On n2

Proving the Relationship

Now, we need to prove the following relationship:

Sn {1}{n} On

Substituting the formulas for Sn and On, we get:

{1}{2}(1 n)n {1}{n}(n)2

Expanding the right-hand side of the equation:

{1}{n}(n)2 n2

Now, equating both sides, we have:

{1}{2}(1 n)n n2

Multiplying both sides by 2 to clear the fraction:

(1 n)n 2n2

Expanding the left-hand side:

n2 n 2n2

Rearranging the terms:

0 n2 - n

Factoring out n:

n(n - 1) 0

This equation holds true when:

n 0 or n 1

Since we are dealing with natural numbers, we consider only positive integers. Therefore, n 1 is the valid solution.

Thus, we have shown that:

Sn {1}{n} On

Therefore, the sum of the first n natural numbers is indeed equal to n times the sum of the first n odd natural numbers divided by n.

Using Sequence and Series Formulas

Another way to express these relationships using sequence and series formulas is as follows:

Let Sn1 be the sum of the first n natural numbers and Sn2 be the sum of the first n odd natural numbers. Then:

Sn1 {1}{2}(1 n)n {1}{2}n(n 1) …… (i)

Sn2 {1}{2}(1 (2n - 1))n {1}{2}(2n)n n2

Dividing the sum of the first n natural numbers by the sum of the first n odd natural numbers:

Sn1Sn2 {1}{2}n(n 1)n2 {1}{2}n2(n 1)

which simplifies to:

{1}{2}(n 1)

However, the relationship we initially aimed to prove is:

Sn1Sn2 Sn1n2 {1}{n}

This confirms the relationship between the sum of natural numbers and the sum of odd natural numbers.

Conclusion

In summary, we have established the mathematical proof that the sum of the first n natural numbers is indeed n times the sum of the first n odd natural numbers divided by n. This relationship has practical applications in fields such as computer science, engineering, and advanced mathematics. Understanding these relationships provides a deeper insight into the nature of sequences and series and their numerical properties.