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Strong induction is a variant of induction, in which we assume that the statement holds for all values preceding \(k\). This provides us with more information to use when trying to prove the statement.
Strong mathematical induction takes the principle of induction a step further by allowing us to assume that the statement holds not only for all natural numbers ‘n ≥1’ but also for (n + 1) or (n+1)th iteration.
Using strong induction, you assume that the statement is true for all $m<n$ (at least your base case) and prove the statement for $n$. In practice, one may just always use strong induction (even if you only need to know that the statement is true for $n-1$).
Equipped with this observation, Bob saw clearly that the strong principle of induction was enough to prove that \(f(n)=2n+1\) for all \(n \geq 1\). So he could power down his computer and enjoy his coffee.
With simple induction you use "if $p(k)$ is true then $p(k+1)$ is true" while in strong induction you use "if $p(i)$ is true for all $i$ less than or equal to $k$ then $p(k+1)$ is true", where $p(k)$ is some statement depending on the positive integer $k$.
Induction also lets us prove theorems about integers n ≥ b for b ∈ Z . Adjust all parts of the proof to use n ≥ b instead of n ≥ 0 . Strong induction lets us assume a stronger inductive hypothesis. This makes some proofs easier.
Strong Induction. The only change in the structure is to the induction assumption. 🔗. In regular induction, we use that we know the statement holds for k to get that it holds for . k + 1. Strong induction is useful when we need to use some smaller case (not just k) to get the statement for . k + 1. 🔗.
There is a form of mathematical induction called strong induction (also called complete induction or course-of-values induction) in which the inductive step requires showing that P (k+1) assuming that P (0),…,P (k) hold. In practice, we only need the results for a handful of previous values.
Strong induction is useful when the result for n = k−1 depends on the result for some smaller value of n, but it’s not the immediately previous value (k). Here’s a classic example:
Strong induction is a type of proof closely related to simple induction. As in simple induction, we have a statement \(P(n)\) about the whole number \(n\), and we want to prove that \(P(n)\) is true for every value of \(n\). To prove this using strong induction, we do the following: The base case.