Understanding the Sum of Cubes of Odd Numbers

In this article, we explore the fascinating properties of the sum of cubes of odd numbers. We will delve into the mathematical series and derive a general formula for the sum of cubes of odd numbers, and then apply this formula to a specific case to find the first 4 digits of such a sum.

Formulation and Simplification

Let's consider the sum of the cubes of the first n odd numbers. We define the sum as:

General Expression

The sum of cubes of all odd integers from 1 to n can be expressed as:

S 13 23 33 … n3

When we want to sum the cubes of all odd integers from 1 to n and subtract the cubes of even integers from 2 to n-1, we can use the following steps:

Extracting the Formula

The sum of cubes of all integers from 1 to n is given by:

(frac{n^2(1^2 2^2 3^2 ... n^2)}{4})

The sum of cubes of even integers from 2 to n-1 can be simplified as:

(8 times frac{(1^2 2^2 3^2 ... (n-1)^2/2)} )

This simplifies to:

(frac{(n-1)^2}{4} times frac{(1^2 2^2 3^2 ... (n-1)^2/2)} )

Combining these, the final sum for odd integers is:

(frac{n^2(1^2 2^2 3^2 ... n^2)}{4} - 8 times frac{(n-1)^2/4 times (1^2 2^2 3^2 ... (n-1)^2/2)})

This simplifies further to:

(frac{n^2(4(n^2 - n 1))/8 - 4[(n-1)(2(n-1) 1)]/8 (n^2 1)/8})

For n 199, we get:

(frac{199^2 times 40000 - 19602}{8} 199990000)

Generalization to First 100 Odd Numbers

For the first 100 odd numbers, we can use the formula for the sum of the cubes of the first n odd numbers:

General Formula

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

(left(frac{n(2n - 1)}{2}right)^2)

For n 100, we calculate:

(frac{100(2 times 100 - 1)}{2} 9950)

And squaring this result:

(9950^2 99002500)

The first four digits of this sum are 9900.

Conclusion

In conclusion, we have derived the formula for the sum of the cubes of odd numbers and applied it to specific cases, including the sum of the first 100 odd numbers. This exploration highlights the elegant patterns and properties present in mathematical series.