Sum of Odd Squares: Formula and Proof

The sum of the squares of the first n odd numbers is a fascinating topic in mathematics with practical applications. This article will explore the formula for the sum of the squares of the first n odd numbers, as well as provide a detailed proof using mathematical induction.

Formula for the Sum of Odd Squares

The formula for the sum of the squares of the first n odd numbers can be expressed as:

Sum 12 32 52 ... (2n-1)2 (frac{n(2n-1)(2n 1)}{3})

The first n odd numbers are given by the sequence 1, 3, 5, ..., (2n-1).

Explanation and Example

This formula is derived using mathematical induction or combinatorial arguments. To illustrate the formula, let's consider an example.

Example: n 3

For n 3:

12 32 52 1 9 25 35

Using the formula:

Sum (frac{3(2*3-1)(2*3 1)}{3}) (frac{3*5*7}{3}) 35

This confirms that the formula is correct.

Verification Using J Programming Language

The formula can also be verified using the J programming language. Here's a function to calculate the sum of odd squares from 1 to n:

f : [:/[::1 2 i. ]

Let's check it with a few values:

f 1 1 f 5 165 f 20 10660 f 100 1333300

Proof Using Summation Formulas

To prove the formula, we need the following three summation formulas:

Sum of squares of the first n natural numbers: (sum_{k1}^{n} k^2 frac{n(n 1)(2n 1)}{6}) Sum of the first n natural numbers: (sum_{k1}^{n} k frac{n(n 1)}{2}) Sum of 1's: (sum_{k1}^{n} 1 n)

General Term of the Summation

The general term for the sum of the squares of the first n odd numbers is:

(1^2 3^2 5^2 ... (2n-1)^2 sum_{k1}^{n} (2k-1)^2)

Let S (1^2 3^2 5^2 ... (2n-1)^2)

Therefore:

(S sum_{k1}^{n} (2k-1)^2 sum_{k1}^{n} (4k^2 - 4k 1))

Expanding the summation:

(S sum_{k1}^{n} 4k^2 - sum_{k1}^{n} 4k sum_{k1}^{n} 1)

Using the above three results:

(S 4sum_{k1}^{n} k^2 - 4sum_{k1}^{n} k sum_{k1}^{n} 1)

Substituting the known summation formulas:

(S 4 cdot frac{n(n 1)(2n 1)}{6} - 4 cdot frac{n(n 1)}{2} n)

Further simplification:

(S frac{n(2n-1)(2n 1)}{3} - n(2n-1))

Therefore:

(S frac{n(2n-1)(2n 1)}{3} - n)

Finally, simplifying further:

(S frac{n(2n-1)(2n 1)}{3})

Generalization to Even Squares

We can also derive the sum of the squares of the first n even numbers. The formula for the sum of the squares of the first n even numbers is:

(S_{even} 4sum_{k1}^{n} k^2 - 4n)

If n is even, the formula gives the sum of the even squares up to n2.

Conclusion

The formula for the sum of the squares of the first n odd numbers is a useful tool in mathematics. By verifying the formula with examples and proving it using summation formulas, we can confidently use it in various mathematical applications.