[JEE Advanced 1984] For a natural number $p$, prove that $p^{n+1}+(p+1)^{2n-1}$ is divisible by $p^2+p+1$ for every positive integer $n$.
For a natural number $p$, prove that $p^{n+1}+(p+1)^{2n-1}$ is divisible by $p^2+p+1$ for every positive integer $n$.
1 Answer
VAVidaara Admin
✓ Vidaara Team
✓ Accepted
· 1mo ago
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Answer: Proved.
Induction on $n$: the step uses $(p+1)^2=p^2+2p+1\equiv p\pmod{p^2+p+1}$, so $(p+1)^{2n-1}$ folds neatly into the divisibility relation.
JEE Advanced 1984 · Binomial Theorem — verified solution by the Vidaara Team.
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