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$$ \sum_{k=1}^n k^4 = {(6n^5+15n^4+10n^3-n) \over 30} $$

How can this be proven using mathematical induction? My teacher isn't any help, he just tells me to think about it, but I've read the textbook again and again, and there isn't much in it that would help me prove this statement.

*Sorry to ask multiple questions on the site in such a short time, but I'm just quite desperate for help and I don't have anyone to go to for it. My teacher barely speaks English, and with my poor hearing I can hardly follow the class and I need to pass this exam desperately.

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    I think your instructor is right. Every induction proof is almost identical. Just follow the general procedure.2012-12-10
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    "Every induction proof is almost identical" Hehe...2012-12-10
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    I would say that the generalization that "every induction proof is identical" is a very poor generalization, and still doesn't help much with learning how to think about these kinds of problems. I'm guessing that what you've done already and the answers provided you have a template for the proof, and now mainly aren't sure of the inner mechanics of this type of problem. In the real world these mechanics are the primary area variation of inductive proofs themselves. In your class (it would seem) practicing polynomial arithmetic is likely going to be a big push to understanding these mechanics.2012-12-10
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    What have you tried so far? Have you checked the base case (when $n = 1$)? Have you written out the inductive hypothesis (suppose the identity holds for $n$). Have you then written out what you are trying to prove (what does the identity say for $n+1$?)? How can you prove this using your inductive hypothesis?2012-12-10

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