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How to prove that Sum[2 i, {i, 1, n}] == n^2 + n

(** This is strange, I posted this in reply to a question, and it appears to have opened a new thread. Also, I cannot find the question in the community anymore. Not sure what happened. Sorry. The question was how to prove that Sum[2 i, {i, 1, n}] == n^2 + n **)

Hi there,

that looks very much like a standard homework problem. In fact Mathematica can do that in one step:

Simplify[Sum[2 i, {i, 1, n}] == n^2 + n]

evaluates to True, so it is correct. But you are probably asked to write the proof in a more standard form. So, for a proof by induction you need to check the statement for a starting value, here $n=1$.

Left hand side:

Sum[2 i, {i, 1, n}] /. n -> 1

gives 2.

Right hand side:

n^2 + n /. n -> 1

gives 2 as well.

Next you need to show that if statement is true for any $n$ then it must be true for $n+1$, too. So,

Simplify[Sum[2 i, {i, 1, n}] + 2 (n + 1)]

needs to equal

Simplify[(n^2 + n) /. n -> n + 1]

evaluating the two last commands gives both times $2+3 n + n^2$, so that they are equal. Note that this is mathematically speaking a bit dodgy because Mathematica already knows the result of the sum:

Expand[Sum[2 i, {i, 1, n}]]

Internally, Mathematica will probably not do the same steps of the argument that you would do on a sheet of paper. WRI are however very interested in automated theorem proving, which you can see in this blog entry.



POSTED BY: Marco Thiel
7 Replies

I stand corrected, it is apparently Theorema 2. As for attendee lists, I think we only mention external attendees.

POSTED BY: Daniel Lichtblau

I believe it was version 3 that came out. Version 2 is circa 2010 or so.

POSTED BY: Daniel Lichtblau

Oh, that is great. I cannot find a link to Theorema 3 yet. (I cannot even find any mention of Theorema 2 before 2012!) Did you attend the presentation at the Wolfram Technology conference? You are not listed as attendee...

The only thing I could find was a talk on Thursday at 5:30 pm which was announced to be on Theorema 2:

Browsing through the program of The WTC, I feel that I missed out on lots of cool stuff. If you can point me towards a download for Theorema 3, I would be very grateful for that.

Cheers and thanks,


POSTED BY: Marco Thiel

Wolfgang Windsteiger did a presentation on Theorema 3 at the Wolfram tgechnology Conference yesterday. It does do inductive proofs of the sort described above.

POSTED BY: Daniel Lichtblau

That's interesting. I'd love to have attended. Theorema 3 is not out yet, is it? Theorema 2 came out in June, if I am not mistaken. I do have Theorema 2.0.0


POSTED BY: Marco Thiel

Was it possibly this thread:

Same expression but a different question about it.... ;-)

POSTED BY: David Reiss

No it was not that one. The question was just about the inductive proof of the

Sum[2 i, {i, 1, n} == n^2 + n

statement. It is really strange, unless I have serious hallucinations - not unheard of at the end of the working week - there was a question about using Mathematica to prove the statement above. It might have been from the same person as the thread you mention.

Anyway, I am not really familiar with automatic theorem proving, which is certainly not the main interest of the person who asked the question in the first place. There are approaches such as Theorema described in Buchberger's article. Do you have any experience with this type of software package? I recently attended a conference on automatic theorem proving which some interesting approaches, and it seems that Stephen Wolfram is also exploring ways of how Mathematica can be used for theorem proving, but he is coming from a different direction, I guess, a direction that might be more appealing to mathematicians.



POSTED BY: Marco Thiel
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