1. What is Sum of Cubes of First N Odd Numbers?

The sum of cubes of first n odd numbers is a mathematical series that calculates the sum of cubes of the first n odd natural numbers (1³ + 3³ + 5³ + ... + (2n-1)³).

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( n \) — Number of terms
• \( S_n \) — Sum of cubes of first n odd numbers

Explanation: This formula provides a direct way to calculate the sum without having to compute each term individually.

3. Mathematical Derivation

Details: The formula is derived from the mathematical pattern observed in the sum of cubes of odd numbers. For n terms, the sum equals n squared multiplied by (2n squared minus 1).

4. Using the Calculator

Tips: Enter a positive integer value for n to calculate the sum of cubes of the first n odd numbers.

5. Frequently Asked Questions (FAQ)

Q1: What are the first few terms of this series?
A: For n=1: 1³ = 1; n=2: 1³ + 3³ = 28; n=3: 1³ + 3³ + 5³ = 153; n=4: 1³ + 3³ + 5³ + 7³ = 496.

Q2: How is this formula derived?
A: The formula can be derived using mathematical induction or by observing the pattern in the sums of cubes of odd numbers.

Q3: What is the difference between sum of cubes of odd numbers and even numbers?
A: The sum of cubes of first n even numbers follows a different pattern: 2³(n(n+1)/2)² = 2n²(n+1)².

Q4: Can this formula be used for fractional values of n?
A: No, n must be a positive integer since it represents the count of terms in the series.

Q5: What are some practical applications of this formula?
A: This formula is primarily used in mathematical theory, number theory problems, and in various mathematical competitions and puzzles.