[JEE Advanced 1992] let p 3 be an integer and alpha beta the roots of x 2
Let $p\ge3$ be an integer and $\alpha,\beta$ the roots of $x^2-(p+1)x+1=0$. Using induction show $\alpha^n+\beta^n$ is (i) an integer and (ii) not divisible by $p$.
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· 2d ago
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Answer: Proved.
Let $u_n=\alpha^n+\beta^n$. Then $u_n=(p+1)u_{n-1}-u_{n-2}$ with $u_0=2,u_1=p+1$ — integers. Modulo $p$, $u_n\equiv u_{n-1}-u_{n-2}$ giving the cycle $2,1,-1,-2,-1,1,\dots$, never $\equiv0$.
JEE Advanced 1992 · Binomial Theorem — verified solution by the Vidaara Team.
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