Sum of Consecutive Cubes Calculator

What is the Sum of Consecutive Cubes?

The sum of consecutive cubes is like adding up the volumes of bigger and bigger cubes! Imagine you have a set of cube blocks, starting with a 1x1x1 cube, then a 2x2x2 cube, then a 3x3x3 cube, and so on. If you add up the volumes of all these cubes, you get the sum of consecutive cubes!

How to Calculate the Sum of Consecutive Cubes

To find the sum of consecutive cubes, we add up the cubes of all whole numbers from 1 to the number we choose. It's like counting how many little cube units are in all our blocks put together!

Formula

The formula for the sum of consecutive cubes is:

\[ S_n = \left(\frac{n(n+1)}{2}\right)^2 \]

Where:

• \(S_n\) is the sum of cubes from 1³ to n³
• \(n\) is the last number in our sequence

Calculation Steps

1. Choose the last number (n) in your sequence
2. Add 1 to your number (n+1)
3. Multiply n by (n+1)
4. Divide the result by 2
5. Square the result (multiply it by itself)
6. The answer is your sum of consecutive cubes!

Example and Visual Representation

Let's find the sum of cubes from 1³ to 4³:

Using our formula: \(S_4 = \left(\frac{4(4+1)}{2}\right)^2 = \left(\frac{4 × 5}{2}\right)^2 = 10^2 = 100\)

Let's check by adding: 1³ + 2³ + 3³ + 4³ = 1 + 8 + 27 + 64 = 100

Here's a picture to show what this looks like:

In this picture, each colored cube represents a cubed number. The red cube is 1³, the blue cube is 2³, the green cube is 3³, and the purple cube is 4³. When we add up all these volumes, we get 100 cubic units!