Exercise · ZW-STRONG-INDUCTION-002 · ★☆☆ Basic
An inductive step proves \(P(k+1)\) using \(P(k)\) and \(P(k-1)\). At least how many base cases are needed?
Choose one option.
Show answer
- A. Two
Explanation
The step uses two earlier values, so for \(k+1=3\) both \(P(1)\) and \(P(2)\) must be checked by hand.
Review the concept: Strong induction
Strong induction proves \(P(n)\) for every \(n\ge n_0\): after checking the base case \(P(n_0)\), one proves \(P(k+1)\) assuming \(P(j)\) for all \(j\) with \(n_0\le j\le k\).
\[\big[P(n_0)\wedge\forall k\ge n_0\,(P(n_0)\wedge\cdots\wedge P(k)\Rightarrow P(k+1))\big]\Rightarrow\forall n\ge n_0\,P(n)\]
More practice problems: strong induction
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