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I recently answered a question and did much more than the question asked.

The question stated a recurrence for a sequence $a_n$ and asked for a proof that $20 < a_{200} < 24$.

I found an asymptotic formula for all $a_n$ in the form $$a_n = \sqrt{2n + \frac{\ln (n)}{2} + O(1)} = \sqrt{2n}+ \frac{\ln (n)}{4\sqrt{2n}}+O(\frac1{\sqrt{n}}) $$ and verified it numerically.

Should this much more exact result be made a separate question?

If not, is there anything else I should do?

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    $\begingroup$ Lot's of people go way beyond the scope of the question. Often times just to help the OP understand things better, but they certainly answer more than the question. So I guess there's nothing to do, just let it be. $\endgroup$
    – Git Gud
    Sep 10, 2013 at 22:40
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    $\begingroup$ Of course, unless you give a bound on the constant for the big-O estimate, that estimate does not really show that $20\lt a_{200}\lt24$. $\endgroup$
    – robjohn Mod
    Sep 10, 2013 at 23:49
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    $\begingroup$ I would like it if going above-and-beyond the question were actively discouraged for questions with the homework tag. There was a time or two on this site where I asked for an unsticking or clarification on a homework problem, and the answerer got carried away and solved the whole thing for me. $\endgroup$
    – GMB
    Sep 11, 2013 at 9:12
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    $\begingroup$ How does this answer do more than asked? It doesn't actually answer the question asked. $\endgroup$
    – copper.hat
    Sep 14, 2013 at 3:45

1 Answer 1

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There are a two ways this can happen:

  1. You first give a canonical solution, then extend it to something better / stronger. (Alternatively, if an answer has already been given containing the canonical solution, you can just post the extended part.)

  2. You prove a more general proposition and then show that their question is a corollary.

In most cases, the first way is best. You want to give them "the answer and more" - make sure it's not just the "and more." It seems like you used this first method and did so correctly.

The second way is best for addressing misconceptions in the question (like pointing out that a part of the hypothesis is superfluous, for example). It's also good for when you don't know any simpler way to solve the problem.

What you don't want to do is give a proof by nuke, then not explain what's going on for the benefit of lower level users. Even if the asker understands why the larger case implies the smaller one, others might not. So, it's good to at least give a nod either before or after your extension for people who may only be capable of understanding the simplest way of getting a solution.

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