Sum of Consecutive Squares Calculator

What is the Sum of Consecutive Squares?

The sum of consecutive squares is like adding up the areas of bigger and bigger squares! Imagine you have a set of square tiles, starting with a 1x1 tile, then a 2x2 tile, then a 3x3 tile, and so on. If you add up the areas of all these tiles, you get the sum of consecutive squares!

How to Calculate the Sum of Consecutive Squares

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

Formula

The formula for the sum of consecutive squares is:

\[ S_n = \frac{n(n+1)(2n+1)}{6} \]

Where:

• \(S_n\) is the sum of squares from 1² to n²
• \(n\) 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 your number by 2 and add 1 (2n+1)
4. Multiply all these numbers together: n × (n+1) × (2n+1)
5. Divide the result by 6
6. The answer is your sum of consecutive squares!

Example and Visual Representation

Let's find the sum of squares from 1² to 4²:

Using our formula: \(S_4 = \frac{4(4+1)(2×4+1)}{6} = \frac{4 × 5 × 9}{6} = 30\)

Let's check by adding: 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30

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

In this picture, each colored square represents a squared number. The red square is 1², the blue square is 2², the green square is 3², and the purple square is 4². When we add up all these areas, we get 30 square units!