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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.