Sum of the first n cubes

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In summary, the conversation is discussing how to derive the formula for the sum of the first n cubes by using the formula for the sum of the first n integers and the method of differences. The formula (n+1)^4 - n^4 is expanded using the Binomial theorem and then both sides are summed to get the formula for the sum of the first n cubes.
  • #1
ehrenfest
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Homework Statement


Can someone help me derive the formula for the sum of the first n cubes from the formula for the sum of the first n integers that elucidates the reason why the former is the square of the latter?


Homework Equations





The Attempt at a Solution

 
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  • #2
Well you could consider the sum from n=1 to N of (n+1)^4-n^4 and then expand using the method of differences
 
  • #3
You mean use the fact that it telescopes?

Then you just get N^4-1? How does that help?
 
  • #4
[tex](n+1)^4 - n^4 = 4n^3 + 6n^2 + 4n + 1[/tex] by the Binomial theorem.

Sum BOTH sides, and the LHS should get (N+1)^4 actually. Do you see how the RHS will be expressions in terms of the first N integers to the power of 3, 2, 1 and 0? You already know the cases for 0, 1 and 2, now you can work out 3.
 

What is the "Sum of the first n cubes"?

The "Sum of the first n cubes" refers to the sum of all the cubes of the first n natural numbers. It can also be written as Σn^3 = 1^3 + 2^3 + 3^3 + ... + n^3.

Why is the "Sum of the first n cubes" important?

The "Sum of the first n cubes" is important because it is a fundamental concept in mathematics and has various applications in fields such as physics, engineering, and computer science. It is also used in the study of number theory and can help in understanding patterns and relationships between numbers.

How do you calculate the "Sum of the first n cubes"?

The "Sum of the first n cubes" can be calculated using the formula Σn^3 = (n(n+1)/2)^2, where n is the number of terms. This formula is known as Faulhaber's formula and can be derived using mathematical induction.

What are some real-life examples of the "Sum of the first n cubes"?

The "Sum of the first n cubes" can be found in various real-life scenarios, such as the calculation of the volume of a cube or the total displacement of an object moving at a constant acceleration. It can also be used to find the total energy of a system in physics and the number of combinations in a Rubik's cube.

Are there any interesting patterns in the "Sum of the first n cubes"?

Yes, there are several interesting patterns in the "Sum of the first n cubes". For example, the sum of the first n odd cubes is equal to the square of the sum of the first n natural numbers. Also, the sum of the first n cubes is always a perfect square. These patterns can be explored further in the study of number theory.

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