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Mathematical Induction

Source: 1992 National High School Mathematics League, Exam One, Problem 15

February 27, 2020
induction

Problem Statement

nn is a natural number, fn(x)=xn+1−x−n−1x−x−1(x≠0,±1)f_n(x)=\frac{x^{n+1}-x^{-n-1}}{x-x^{-1}}(x\neq0,\pm1), let y=x+1xy=x+\frac{1}{x}. (a) Prove that fn+1(x)=yfn(x)−fn−1(x)f_{n+1}(x)=yf_n(x)-f_{n-1}(x) (b) Prove with mathematical induction: fn(x)={yn−Cn−11yn−2+⋯+(−1)iCn−iiyn−2i+⋯+(−1)n2(i=1,2,⋯ ,n2,n is even)yn−Cn−11yn−2+⋯+(−1)iCn−iiyn−2i+⋯+(−1)n−12Cn+12n−12y(i=1,2,⋯ ,n−12,n is odd)f_n(x)=\begin{cases} y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n}{2}}(i=1,2,\cdots,\frac{n}{2},n\text{ is even})\\ y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n-1}{2}}\text{C}_{\frac{n+1}{2}}^{\frac{n-1}{2}}y(i=1,2,\cdots,\frac{n-1}{2},n\text{ is odd}) \end{cases}.
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