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Math and Logic

Series of Squares

Humans are a visual species, which is why many of us have trouble with mathematical abstractions. But that also means we can visualize those abstractions, unlocking a powerful tool to quickly solve complex problems.

Algebraic simplification is one route to solving today's problem, but a quick geometric visualization can bypass almost all the algebra. To start, visualize the squared numbers as square arrays of dots:

We can see what happens when a square array is subtracted from another that is slightly larger.

So, \(9^2 - 8^2 = 9+8.\)

This might seem familiar if you've seen the difference of squares formula before: \[\begin{align} a^2 - b^2 &= (a+b)(a-b) \\ 9^2 - 8^2 &= (9+8)(1). \end{align}\] However, today's problem is a lengthy, alternating sum of squares, a longer version of \[9^2 - 8^2 + 7^2 - 6^2 + 5^2 - 4^2 + 3^2 - 2^2 + 1^2.\] You can simplify this algebraically if you want... or, if we represent the full series of addition and subtraction visually, it's possible to see the simplification visually. The light pink circles are those being "removed" by each subtraction in the sequence.

Lastly, we can rearrange the remaining dots, which represent the total sum of the series. This arrangement can help us determine a simpler calculation to find the total number of dots as this triangle fills exactly half of a \(9 \times 10\) rectangle cells.

Today's Problem

\[\begin{align} 2^2 - 1^2 &= 3\\ 3^2 - 2^2 + 1^2 &= 6\\ 4^2 - 3^2 + 2^2 - 1^2 &= 10\\ 5^2 - 4^2 + 3^2 - 2^2 + 1^2 &= 15\\ \end{align}\]

Calculate the value of \(x:\) \[100^2 - 99^2 + \cdots + 2^2 - 1^2\ = x.\]

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