Prove or disprove n² + 100n is Θ(n²)

[bob]

I see you questioning whether that polynomial stays in the same growth class as plain n squared. It does hold up under the definition. I worked through the bounds myself last week. You can find constants that sandwich it nicely for big enough inputs. Perhaps we start by noting the extra term fades away relatively.

And the main part stays dominant always. I recall proving similar cases back in my early projects. You probably spot how the linear piece gets swallowed. But we need explicit numbers to lock it down tight. Or else the proof falls flat without them.

Now consider picking a lower multiplier of one. That works because the whole thing exceeds n squared outright. I know you agree it never dips below. You add positive stuff so it grows at least that fast. Maybe test small values first to see the pattern hold.

Then for the upper side I choose two as the cap. You see after n passes one hundred the extra hundred n stays under n squared. I verified that by rearranging the inequality myself. But it clicks once you solve for when n beats one hundred. Or try n at two hundred to confirm the gap widens.

Also the sandwich stays valid onward from there. I like how simple the choice turns out. You avoid hunting fancy multipliers when basic ones suffice. Perhaps graph it mentally to watch the curves align asymptotically. But the limit of the ratio approaches one anyway.

You push further by dividing both sides by n squared. I end up with one plus one over n going to one. That confirms the theta relation without doubt. Or recall how we skip lower terms in practice. Maybe you tried this on other quadratics before.

Now the proof stands complete once constants lock in. I chose c one equals one and c two equals two. You need n zero at one hundred for safety. But adjust it higher if you want stricter room. Or experiment with your own values next time.

Also think about what happens if the linear coefficient grew huge. I wonder if it would still qualify then. You might stretch n zero farther out. Perhaps the same idea scales but constants change. But here it stays straightforward.

I keep coming back to how this guides picking efficient code paths. You weigh similar growth rates when comparing options. Or maybe compare against cubic cases to contrast. Now the point lands that they match in big picture scaling.

You ask why bother with such checks at all. I answer it saves time spotting equivalent runtimes early. But the math feels solid once practiced. Perhaps share your own example next round. Or we could tweak the expression slightly for variety.

The idea extends to many other polynomials too. I proved a few variants during recent reviews. You gain intuition after handling several. Maybe the hundred n term tricks newcomers sometimes. But it never breaks the theta class here.

Also remember limits help confirm without guessing constants. I compute the ratio limit equals one. You see it squeezes between positives. Or the definition gets satisfied directly. Now it all ties together cleanly.

I think you got the flow by now. You apply the same steps elsewhere easily. But always verify the n zero point holds. Perhaps run a quick loop in your mind for checks. Or trust the algebra once derived.

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