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1, 4, 9, 16, 25, 36, 49…And now find the difference between consecutive squares:

1 to lớn 4 = 34 to 9 = 59 to lớn 16 = 716 to lớn 25 = 925 to 36 = 11…Huh? The odd numbers are sandwiched between the squares?

Strange, but true. Take some time lớn figure out why — even better, find a reason that would work on a nine-year-old. Go on, I’ll be here.

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Exploring Patterns

We can explain this pattern in a few ways. But the goal is to find a convincing explanation, where we slap our forehands with “ah, that’s why!”. Let’s jump into three explanations, starting with the most intuitive, và see how they help explain the others.

Geometer’s Delight

It’s easy khổng lồ forget that square numbers are, well… square! Try drawing them with pebbles

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Notice anything? How vì we get from one square number khổng lồ the next? Well, we pull out each side (right and bottom) và fill in the corner:

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While at 4 (2×2), we can jump lớn 9 (3×3) with an extension: we add 2 (right) + 2 (bottom) + 1 (corner) = 5. & yep, 2×2 + 5 = 3×3. & when we’re at 3, we get lớn the next square by pulling out the sides and filling in the corner: Indeed, 3×3 + 3 + 3 + 1 = 16.

Each time, the change is 2 more than before, since we have another side in each direction (right & bottom).

Another neat property: the jump to lớn the next square is always odd since we change by “2n + 1″ (2n must be even, so 2n + 1 is odd). Because the change is odd, it means the squares must cycle even, odd, even, odd…

And wait! That makes sense because the integers themselves cycle even, odd, even odd… after all, a square keeps the “evenness” of the root number (even * even = even, odd * odd = odd).

Funny how much insight is hiding inside a simple pattern. (I hotline this technique “geometry” but that’s probably not correct — it’s just visualizing numbers).

An Algebraist’s Epiphany

Drawing squares with pebbles? What is this, ancient Greece? No, the modern student might argue this:

We have two consecutive numbers, n and (n+1)Their squares are n2 and (n+1)2The difference is (n+1)2 – n2 = (n2+ 2n + 1) – n2 = 2n + 1

For example, if n=2, then n2=4. & the difference to lớn the next square is thus (2n + 1) = 5.

Indeed, we found the same geometric formula. But is an algebraic manipulation satisfying? khổng lồ me, it’s a bit sterile & doesn’t have that same “aha!” forehead slap. But, it’s another tool, và when we combine it with the geometry the insight gets deeper.

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Calculus Madness

Calculus students may think: “Dear fellows, we’re examining the curious sequence of the squares, f(x) = x^2. The derivative shall reveal the difference between successive elements”.

And deriving f(x) = x^2 we get:

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Close, but not quite! Where is the missing +1?

Let’s step back. Calculus explores smooth, continuous changes — not the “jumpy” sequence we’ve taken from 22 to 32 (how’d we skip from 2 lớn 3 without visiting 2.5 or 2.00001 first?).

But don’t đại bại hope. Calculus has algebraic roots, và the +1 is hidden. Let’s dust off the definition of the derivative:

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Forget about the limits for now — focus on what it means (the feeling, the love, the connection!). The derivative is telling us “compare the before and after, & divide by the change you put in”. If we compare the “before và after” for f(x) = x^2, and call our change “dx” we get:

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Now we’re getting somewhere. The derivative is deep, but focus on the big picture — it’s telling us the “bang for the buck” when we change our position from “x” to “x + dx”. For each unit of “dx” we go, our result will change by 2x + dx.

For example, if we pick a “dx” of 1 (like moving from 3 khổng lồ 4), the derivative says “Ok, for every unit you go, the đầu ra changes by 2x + dx (2x + 1, in this case), where x is your original starting position and dx is the total amount you moved”. Let’s try it out:

Going from 32 to 42 would mean:

x = 3, dx = 1change per unit input: 2x + dx = 6 + 1 = 7amount of change: dx = 1expected change: 7 * 1 = 7actual change: 42 – 32 = 16 – 9 = 7

We predicted a change of 7, và got a change of 7 — it worked! và we can change “dx” as much as we like. Let’s jump from 32 to lớn 52:

x = 3, dx = 2change per unit input: 2x + dx = 6 + 2 = 8number of changes: dx = 2total expected change: 8 * 2 = 16actual change: 52 – 32 = 25 – 9 = 16

Whoa! The equation worked (I was surprised too). Not only can we jump a boring “+1″ from 32 to 42, we could even go from 32 to lớn 102 if we wanted!

Sure, we could have figured that out with algebra — but with our calculus hat, we started thinking about arbitrary amounts of change, not just +1. We took our rate và scaled it out, just lượt thích distance = rate * time (going 50mph doesn’t mean you can only travel for 1 hour, right? Why should 2x + dx only apply for one interval?).

My pedant-o-meter is buzzing, so remember the giant caveat: Calculus is about the micro scale. The derivative “wants” us lớn explore changes that happen over tiny intervals (we went from 3 to lớn 4 without visiting 3.000000001 first!). But don’t be bullied — we got the idea of exploring an arbitrary interval “dx”, and dagnabbit, we ran with it. We’ll save tiny increments for another day.

Lessons Learned

Exploring the squares gave me several insights:

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Seemingly simple patterns (1, 4, 9, 16…) can be examined with several tools, to get new insights for each. I had completely forgotten that the ideas behind calculus (x going to lớn x + dx) could help investigate discrete sequences.It’s all too easy khổng lồ sandbox a mathematical tool, like geometry, & think it can’t shed light into higher levels (the geometric pictures really help the algebra, especially the +1, pop). Even with calculus, we’re used khổng lồ relegating it to tiny changes — why not let dx stay large?Analogies work on multiple levels. It’s clear that the squares và the odds are intertwined — starting with one set, you can figure out the other. Calculus expands this relationship, letting us jump back và forth between the integral và derivative.

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As we learn new techniques, don’t forget lớn apply them to the lessons of old. Happy math.

Appendix: The Cubes!

I can’t help myself: we studied the squares, now how about the cubes?

1, 8, 27, 64…

How vì chưng they change? Imagine growing a cube (made of pebbles!) khổng lồ a larger & larger kích cỡ — how does the volume change?