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Possible Duplicate:
Prove existence of a real root.

If $a_0$+$\frac{a_1}{2}$+$\frac{a_2}{3}+\ldots+\frac{a_n}{n+1}=0$, how to prove ${{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+\cdots +{{a}_{n}}{{x}^{n}}=0$ has at least one real root in $(0,1)$.

I know constructor $f(x)={{a}_{0}}x+\frac{{{a}_{1}}}{2}{{x}^{2}}+\frac{{{a}_{2}}}{3}{{x}^{3}}+\cdots +\frac{{{a}_{n}}}{n+1}{{x}^{n+1}}$, and then use the Mean Value Theorem.

I want to know whether we can use mathematical induction to prove, obviously $n = 1$ proposition holds.

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    "I know constructor"? What do you mean? Anyway, have you tried to prove the induction step? Does it work?2012-10-08
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    This is a natural number proposition.So I wanted to try to use mathematical induction.I made some attempts, without success, to seek help.Thank you2012-10-08
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    Perhaps possible.. but not easy.. WHY do you insist on induction?2012-10-08
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    There is no special intention.Just suddenly thought, because the classical solution of this problem is the intermediate value theorem, I want to try the other solution is also very interesting.Thank you2012-10-08
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    Let me know if you find one..2012-10-08
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    Is this really a duplicate? The OP seems to ask about possibility of the proof using induction. (Although I have doubts that there is a simple proof using induction.)2012-10-09

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