Competition problem — Number theory

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Number theory Difficulty 6.1 National olympiad Prove it

1. Let $k_{0}$ be a given integer, and $P(n)$ be a property or proposition concerning the integer $n$ . If
(i) $P\left(k_{0}\right)$ holds when $n=k_{0}$ ;
(ii) $P(n)$ holding implies that $P(n+1)$ holds,

then $P(n)$ holds for all integers $n \geqslant k_{0}$ .

Solution

1. Make a variable substitution: $n=m+k_{0}-1 . P(n)=P\left(m+k_{0}-1\right)=P^{*}(m)$ . Apply Theorem 1 of §1 to $P^{*}(m)$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.